What's the minimum number of $2$s needed to write a positive integer?
I've been silly; we don't need to work with iterated logarithms. We can get a logarithmic upper bound by using the binary expansion slightly more cleverly. Namely, we can always write $n = 2n' + \left( n \bmod 2 \right)$, so either $2k = 2(k)$ or $2k+1 = 2(k) + \frac 22$, which gives
$$f(2k) \le f(k) + 1$$ $$f(2k+1) \le f(k) + 3.$$
Iterating these bounds gives
$$\boxed{ f(n) \le 3 \lceil \log_2 n \rceil - 1 \approx 4.32 \log n }$$
which corresponds to writing $n$ as $d_0 + 2(d_1 + 2(d_2 + \dots)))$ where $d_i$ are the binary digits of $n$. This uses only addition, multiplication and division and lots of optimizations are possible. This gives $f(14885392687) \le 3 \cdot 33 + 2 = 101$ which is at least within a factor of $3$ of the explicit result.
As an example of a possible optimization, we can improve the bound by working in base $22$, which gives
$$f(n) \le \left( 2 + g(21) \right) \lceil \log_{22} n \rceil.$$
My computations give $g(21) \le 5$ (the maximum value of $5$ is attained for $n = 7, 15, 17, 19$, at least if I'm not mistaken), so
$$\boxed{ f(n) \le 7 \lceil \log_{22} n \rceil \approx 2.26 \log n }$$
which is almost twice as good! This gives $f(14885392687) \le 56$ which still doesn't quite match the explicit result. Using subtraction we can replace $g(21)$ above by $g(10)$ but since $g(10) = 5$ also this doesn't actually help in this case.
We can write down a logarithmic lower bound on $g$ by writing down an exponential upper bound on the number $N(k)$ of possible expressions involving $k$ twos. (At least one number between $1$ and $N(k)+1$ can't be represented using $k$ twos, so $g(N(k) + 1) \ge k+1$.) We can do a more precise count than the following but this will suffice. An expression involving $k$ twos involves at most $k-1$ operations and at most $k-1$ pairs of parentheses, so altogether is a string of at most $4k-3$ symbols each of which can take the values $2, (, ), +, -, \times, \div$, or exponentiation (note that we don't need a symbol for concatenation). This gives the crude bound $N(k) \le 7^{4k-3}$, so
$$g(7^{4k-3} + 1) \ge k + 1$$
which after a bit of massaging gives
$$\boxed{ g(n) \ge \frac{\lceil \log_7 n \rceil + 3}{4} \approx 0.128 \log n }.$$
This gives $g(14885392687) \ge 4$ which is quite bad! Can anyone do substantially better, possibly after disallowing some of the operations? A lower bound given only addition, multiplication, and exponentiation would already be quite interesting, I think.
Just observation.
For given $d$ ($d\ge 2$) consider "milestone values" $k(d)$: the smallest number, which requires $d$ $\;2$s
(such that all numbers below $k(d)$ require $<d$ $\;2$s).
Example:
for $d=5$ all numbers below $7$ could be expressed via $<5$ $\;2$s, but $7$ requires $5$ $\;2$s:
$$
7 = 2+2+2+2/2 = 2^2+2+2/2 = 2\times 2\times 2 - 2/2.
$$
Therefore, $k(5)=7$.
According to my computations (checking all numbers below $150\;000$), first values for $k(d)$ are:
\begin{array}{|c|c|l|} \hline d & k(d) & example \; of \; expression \\ \hline 2 & 1 & 1=2/2 \\ 3 & 3 & 3 = 2+2/2 \\ 4 & 5 & 5 = 2+2+2/2 \\ 5 & 7 & 7 = 2+2+2+2/2 \\ 6 & 27 & 27 = 3^3 = (2+2/2)^{2+2/2} \\ 7 & 29 & 29 = 22/2 + 22-2-2 \\ 8 & 149 & 149 = (4^4-2)/2+22 = ((2+2)^{2+2}-2)/2+22 \\ 9 & 271 & 271 = 222 + (22+2)\times 2 + 2/2 \\ 10 & 691 & 691 = (22+2)^2 + 222/2 + 2+2 \\ 11 & 1381 & 1381 = (222+2)\times (2+2) + 22^2 + 2/2 \\ 12 & 3493 & 3493 = (222-2-2) \times 2^{2+2} + 2+2+2/2 \\ 13 & 9907 & 9907 = 484\times 20 + 227 = 22^2\times (22-2) + 222 + 2+2+2/2 \\ 14 & 34\:093 & 34\:093 = 2^{16}/2 + 11^3-6 = 2^{(2+2)^2}/2 + (22/2)^{2+2/2}-2-2-2 \\ 15 & 120\:347 & 120\:347 = (222-2/2)^2 + 222^2+22222 \\ 16 & 305\:421 & 305\:421 = \bigl(((22-2) \times (2+2)^2 - 2/2)^2 + 2\times 22 + 2\bigr) \times (2 + 2/2) \\ ... & ... & ... \end{array}
We can observe that for $d>12$ $\;\;$ $\log_2 k(d) > d$, which (probably) can promise that for large enough $n$ one can expect estimation $$ f(n) \le \log_2 n. $$
To know about such decomposition up to number $n$, I create $3$ arrays:
f[n], o1[n], o2[n].
f[n] - keeps the number of $2$s in the shortest decomposition of the $n$;
o1[n] - for keeping of $1$st operand;
o2[n] - for keeping of $2$nd operand.
all the arrays is filled initially by $0$;
and setting manually f[2]=1, o1[2]=2, o2[2]=0.
After that, I loop through $a,b$, where $1 \le a \le b \le n$,
calculate all possible values
$c=a+b$,
$c=b-a$,
$c=b/a$ (if integer),
$c=a*b$ (if not too large),
$c=a^b$ (if not too large),
$c=b^a$ (if not too large).
If calculated value $c$ is new or requires less $2$s than existing one, then I change corresponding array values f[c], o1[c], o2[c].
And repeat this loop while it provides at least one improvement.
Then, we can be more or less confident about decompositions in the range [1 .. n/2].
And here are first few hundreds of these arrays:
n f(n) o1(n) o2(n)
------------------
1 2 2 2
2 1 2 -
3 3 2 1
4 2 2 2
5 4 3 2
6 3 4 2
7 5 4 3
8 3 4 2
9 4 3 2
10 4 8 2
11 3 22 2
12 4 6 2
13 4 11 2
14 4 16 2
15 5 11 4
16 3 4 2
17 5 16 1
18 4 16 2
19 5 20 1
20 3 22 2
21 4 22 1
22 2 22 -
23 4 22 1
24 3 22 2
25 5 5 2
26 4 22 4
27 6 3 3
28 5 14 2
29 7 16 13
30 5 22 8
31 6 20 11
32 4 16 2
33 5 22 11
34 5 32 2
35 6 22 13
36 4 6 2
37 6 36 1
38 5 22 16
39 6 40 1
40 4 20 2
41 6 40 1
42 4 44 2
43 5 44 1
44 3 22 2
45 5 44 1
46 4 44 2
47 6 44 3
48 4 24 2
49 6 7 2
50 5 48 2
51 7 40 11
52 5 26 2
53 7 42 11
54 6 32 22
55 6 44 11
56 6 14 4
57 7 44 13
58 6 36 22
59 7 48 11
60 6 20 3
61 7 62 1
62 5 64 2
63 6 64 1
64 4 6 2
65 6 64 1
66 5 22 3
67 7 44 23
68 6 34 2
69 7 23 3
70 6 48 22
71 7 72 1
72 5 36 2
73 7 72 1
74 6 72 2
75 7 64 11
76 6 38 2
77 7 79 2
78 6 80 2
79 6 81 2
80 5 20 4
81 5 4 3
82 6 80 2
83 6 81 2
84 5 42 2
85 7 81 4
86 5 88 2
87 6 88 1
88 4 22 4
89 6 88 1
90 5 88 2
91 7 88 3
92 5 46 2
93 7 92 1
94 6 92 2
95 7 96 1
96 5 24 4
97 7 96 1
98 6 96 2
99 6 121 22
100 5 10 2
101 6 2222 22
102 6 100 2
103 7 81 22
104 6 26 4
105 7 107 2
106 7 84 22
107 6 109 2
108 6 110 2
109 5 111 2
110 5 220 2
111 4 222 2
112 5 224 2
113 5 111 2
114 6 112 2
115 6 111 4
116 7 58 2
117 6 119 2
118 7 96 22
119 5 121 2
120 6 20 6
121 4 11 2
122 6 121 1
123 5 121 2
124 6 62 2
125 6 121 4
126 6 128 2
127 6 254 2
128 5 64 2
129 6 258 2
130 6 128 2
131 7 109 22
132 5 22 6
133 6 111 22
134 6 132 2
135 7 111 24
136 7 34 4
137 7 121 16
138 7 23 6
139 7 278 2
140 7 70 2
141 7 119 22
142 6 144 2
143 6 121 22
144 5 12 2
145 7 121 24
146 6 144 2
147 7 169 22
148 7 74 2
149 8 127 22
150 7 128 22
151 8 111 40
152 7 38 4
153 8 109 44
154 7 14 11
155 7 111 44
156 7 26 6
157 8 111 46
158 7 79 2
159 8 111 48
160 6 20 8
161 7 322 2
162 6 81 2
163 7 326 2
164 7 82 2
165 7 121 44
166 7 83 2
167 6 169 2
168 6 42 4
169 5 13 2
170 7 168 2
171 6 169 2
172 6 86 2
173 7 169 4
174 6 176 2
175 7 176 1
176 5 22 8
177 7 176 1
178 6 176 2
179 8 176 3
180 6 90 2
181 8 180 1
182 7 180 2
183 8 184 1
184 6 46 4
185 8 121 64
186 7 184 2
187 8 17 11
188 7 94 2
189 7 378 2
190 7 192 2
191 7 169 22
192 6 24 8
193 8 169 24
194 6 196 2
195 7 196 1
196 5 14 2
197 7 196 1
198 6 22 9
199 6 398 2
200 5 222 22
201 6 402 2
202 6 200 2
203 7 201 2
204 7 102 2
205 8 201 4
206 6 222 16
207 8 23 9
208 7 16 13
209 7 211 2
210 7 222 12
211 6 222 11
212 7 214 2
213 7 211 2
214 6 222 8
215 8 211 4
216 6 6 3
217 7 218 1
218 5 220 2
219 6 220 1
220 4 222 2
221 5 222 1
222 3 222 -
223 5 222 1
224 4 222 2
225 6 15 2
226 5 222 4
227 7 222 5
228 6 222 6
229 7 231 2
230 6 222 8
231 6 462 2
232 7 222 10
233 6 222 11
234 6 256 22
235 7 222 13
236 7 220 16
237 7 239 2
238 6 119 2
239 6 241 2
240 5 242 2
241 5 482 2
242 4 484 2
243 5 486 2
244 5 222 22
245 6 243 2
246 6 123 2
247 7 243 4
248 7 62 4
249 8 241 8
250 7 125 2
251 7 253 2
252 6 254 2
253 6 506 2
254 5 256 2
255 6 256 1
256 4 4 4
257 6 256 1
258 5 256 2
259 7 256 3
260 6 256 4
261 8 239 22
262 6 484 222
263 7 241 22
264 6 22 12
265 7 243 22
266 6 222 44
267 7 256 11
268 7 134 2
269 8 256 13
270 7 222 48
271 9 222 49
272 7 256 16
273 8 21 13
274 8 137 2
275 8 25 11
276 7 46 6
277 7 554 2
278 6 256 22
279 8 256 23
280 7 20 14
281 9 241 40
282 8 141 2
283 8 285 2
284 7 142 2
285 7 287 2
286 6 22 13
287 6 574 2
288 5 576 2
289 6 17 2
290 6 288 2
291 7 289 2
292 7 146 2
293 8 289 4
294 8 21 14
295 9 287 8
296 8 74 4
297 8 299 2
298 8 254 44
299 7 598 2
300 7 256 44
301 8 299 2
302 7 324 22
303 8 222 81
304 8 19 16
305 9 222 83
306 7 308 2
307 8 308 1
308 6 22 14
309 8 287 22
310 7 222 88
311 8 289 22
312 7 24 13
313 8 324 11
314 8 222 92
315 8 484 169
316 8 79 4
317 9 196 121
318 7 320 2
319 8 320 1
320 6 20 16
321 8 320 1
322 6 324 2
323 7 324 1
324 5 18 2
325 7 324 1
326 6 324 2
327 8 109 3
328 7 324 4
329 9 218 111
330 7 22 15
331 8 220 111
332 8 83 4
333 7 111 3
334 7 167 2
335 8 222 113
336 7 21 16
337 7 674 2
338 6 169 2
339 7 678 2
340 7 338 2
341 8 220 121
342 7 171 2
343 7 222 121
344 7 86 4
345 8 222 123
346 7 324 22
347 9 222 125
348 7 174 2
349 8 350 1
350 6 352 2
351 7 352 1
352 5 22 16
353 7 352 1
354 6 352 2
355 8 352 3
356 7 178 2
357 8 119 3
358 8 352 6
359 7 361 2
360 7 20 18
361 6 19 2
362 8 360 2
363 7 121 3
364 8 26 14
365 8 361 4
366 8 222 144
367 8 256 111
368 7 23 16
369 8 123 3
370 8 368 2
371 8 373 2
372 8 62 6
373 7 484 111
374 7 22 17
375 8 373 2
376 7 378 2
377 8 256 121
378 6 400 22
379 8 378 1
380 7 378 2
381 9 127 3
382 7 384 2
383 8 361 22
384 6 24 16
385 8 384 1
386 7 384 2
387 8 389 2
388 7 194 2
389 7 400 11
390 7 392 2
391 8 222 169
392 6 196 2
393 8 392 1
394 7 392 2
395 8 396 1
396 6 22 18
397 7 398 1
398 5 400 2
399 6 400 1
400 4 20 2
401 6 400 1
402 5 400 2
403 7 400 3
404 6 400 4
405 8 400 5
406 7 400 6
407 9 37 11
408 7 400 8
409 8 398 11
410 8 400 10
411 7 400 11
412 7 206 2
413 8 400 13
414 8 23 18
415 9 399 16
416 7 26 16
417 8 419 2
418 7 22 19
419 7 441 22
420 7 21 20
421 8 399 22
422 6 400 22
423 8 400 23
424 7 400 24
425 8 441 16
426 7 448 22
427 9 425 2
428 7 214 2
429 8 440 11
430 8 428 2
431 8 433 2
432 7 24 18
433 7 444 11
434 7 436 2
435 8 433 2
436 6 218 2
437 7 439 2
438 6 440 2
439 6 441 2
440 5 22 20
441 5 21 2
442 5 444 2
443 6 441 2
444 4 222 2
445 6 444 1
446 5 444 2
447 7 444 3
448 5 224 2
449 7 448 1
450 6 448 2
451 8 440 11
452 6 226 2
453 8 442 11
454 7 452 2
455 7 444 11
456 7 228 2
457 8 441 16
458 7 460 2
459 8 448 11
460 6 462 2
461 7 462 1
462 5 484 22
463 7 441 22
464 6 462 2
465 8 243 222
466 6 444 22
467 8 444 23
468 6 484 16
469 8 468 1
470 7 448 22
471 7 473 2
472 7 484 12
473 6 484 11
474 7 476 2
475 7 473 2
476 6 484 8
477 8 473 4
478 6 480 2
479 7 480 1
480 5 482 2
481 6 482 1
482 4 484 2
483 5 484 1
484 3 22 2
485 5 484 1
486 4 484 2
487 6 484 3
488 5 484 4
489 7 484 5
490 6 484 6
491 8 480 11
492 6 484 8
493 7 482 11
494 7 484 10
495 6 484 11
496 7 484 12
497 7 484 13
498 7 482 16
499 8 483 16
500 6 484 16
501 8 484 17
502 7 480 22
503 8 481 22
504 6 482 22
505 7 483 22
506 5 484 22
507 7 484 23
508 6 254 2
509 8 484 25
510 6 512 2
511 7 512 1
512 5 9 2
513 7 512 1
514 6 512 2
515 8 512 3
516 6 258 2
517 8 484 33
518 7 516 2
519 9 398 121
520 7 26 20
521 8 400 121
522 8 482 40
523 8 512 11
524 7 262 2
525 7 527 2
526 6 528 2
527 6 529 2
528 5 24 22
529 5 23 2
530 6 528 2
531 6 529 2
532 7 266 2
533 7 529 4
534 7 512 22
535 8 529 6
536 8 134 4
537 8 529 8
538 8 516 22
539 8 528 11
540 8 90 6
541 9 528 13
542 9 320 222
543 9 527 16
544 8 34 16
545 8 529 16
546 8 26 21
547 9 483 64
548 7 484 64
549 8 527 22
550 7 25 22
551 7 529 22
552 7 24 23
553 8 529 24
554 6 576 22
555 7 1110 2
556 7 278 2
557 8 555 2
558 8 554 4
559 9 43 13
560 7 576 16
561 8 1122 2
562 8 560 2
563 8 565 2
564 8 484 80
565 7 576 11
566 8 568 2
567 8 565 2
568 7 576 8
569 9 400 169
570 7 572 2
571 8 572 1
572 6 26 22
573 7 574 1
574 5 576 2
575 6 576 1
576 4 24 2
577 6 576 1
578 5 576 2
579 7 576 3
580 6 576 4
581 8 576 5
582 7 576 6
583 9 361 222
584 7 576 8
585 8 574 11
586 8 576 10
587 7 576 11
588 8 42 14
589 8 576 13
590 8 574 16
591 9 480 111
592 7 576 16
593 8 482 111
594 8 27 22
595 7 484 111
596 7 574 22
597 8 484 113
598 6 576 22
599 8 576 23
600 7 576 24
601 9 480 121
602 8 576 26
603 8 482 121
604 8 302 2
605 7 484 121
606 9 101 6
607 8 484 123
608 8 38 16
609 9 484 125
610 9 482 128
611 9 484 127
612 8 306 2
613 9 444 169
614 8 616 2
615 9 123 5
616 7 28 22
617 9 484 133
618 8 574 44
619 9 575 44
620 7 576 44
621 8 623 2
622 7 400 222
623 7 625 2
624 7 26 24
625 6 5 4
626 8 624 2
627 7 625 2
628 8 484 144
629 8 625 4
630 9 30 21
631 9 625 6
632 8 676 44
633 9 211 3
634 9 632 2
635 9 637 2
636 8 318 2
637 8 1274 2
638 8 640 2
639 9 528 111
640 7 32 20
641 9 400 241
642 8 400 242
643 9 400 243
644 7 322 2
645 8 647 2
646 7 648 2
647 7 1294 2
648 6 324 2
649 7 1298 2
650 7 648 2
651 8 649 2
652 7 326 2
653 8 484 169
654 7 676 22
655 9 484 171
656 8 328 2
657 9 219 3
658 8 660 2
659 8 1318 2
660 7 30 22
661 9 439 222
662 8 440 222
663 8 221 3
664 7 666 2
665 8 666 1
666 6 222 3
667 8 666 1
668 7 666 2
669 8 223 3
670 8 448 222
671 9 649 22
672 7 42 16
673 8 674 1
674 6 676 2
675 7 676 1
676 5 26 2
677 7 676 1
678 6 676 2
679 8 676 3
680 7 676 4
681 9 676 5
682 8 31 22
683 9 441 242
684 8 171 4
685 9 444 241
686 8 343 2
687 8 576 111
688 8 43 16
689 9 576 113
690 9 30 23
691 10 448 243
692 8 346 2
693 9 33 21
694 9 672 22
695 9 473 222
696 8 174 4
697 8 576 121
698 7 676 22
699 9 233 3
700 7 350 2
701 9 700 1
702 7 704 2
703 8 704 1
704 6 32 22
705 8 483 222
706 6 484 222
707 8 484 223
708 7 354 2
709 9 484 225
710 8 484 226
711 9 1111 400
712 8 178 4
713 9 729 16
714 8 119 6
715 9 65 11
716 9 358 2
717 9 239 3
718 8 359 2
719 9 720 1
720 7 36 20
721 8 1442 2
722 7 361 2
723 8 241 3
724 8 482 242
725 8 484 241
726 7 33 22
727 7 729 2
728 8 484 244
729 6 6 3
730 8 729 1
731 7 729 2
732 8 244 3
733 8 729 4
734 8 512 222
735 9 245 3
736 7 46 16
737 9 484 253
738 8 123 6
739 9 483 256
740 7 484 256
741 9 484 257
742 8 484 258
743 10 484 259
744 8 746 2
745 9 576 169
746 7 968 222
747 9 746 1
748 7 34 22
749 9 527 222
750 8 528 222
751 8 529 222
752 8 376 2
753 9 529 224
754 8 756 2
755 9 756 1
756 7 378 2
757 9 756 1
758 8 756 2
759 9 33 23
760 8 38 20
761 10 400 361
762 8 254 3
763 10 109 7
764 8 382 2
765 9 255 3
766 8 768 2
767 9 768 1
768 7 32 24
769 9 768 1
770 8 35 22
771 9 257 3
772 8 386 2
773 9 484 289
774 8 258 3
775 10 484 291
776 8 194 4
777 9 111 7
778 7 800 22
779 9 778 1
780 8 390 2
781 9 782 1
782 7 784 2
783 8 784 1
784 6 28 2
785 8 784 1
786 7 784 2
787 9 676 111
788 8 394 2
789 8 800 11
790 7 792 2
791 8 792 1
792 6 36 22
793 8 792 1
794 7 792 2
795 8 796 1
796 6 398 2
797 8 796 1
798 6 800 2
799 7 800 1
800 5 400 2
801 7 800 1
802 6 800 2
803 8 800 3
804 6 402 2
805 8 804 1
806 7 804 2
807 9 796 11
808 7 404 2
809 9 798 11
810 8 808 2
811 8 800 11
812 8 406 2
813 9 800 13
814 8 37 22
815 9 804 11
816 8 34 24
817 9 576 241
818 8 576 242
819 9 576 243
820 8 798 22
821 9 799 22
822 7 800 22
823 9 800 23
824 8 206 4
825 9 1936 1111
826 8 804 22
827 10 484 343
828 8 36 23
829 10 576 253
830 9 574 256
831 10 277 3
832 8 32 26
833 10 119 7
834 8 836 2
835 9 836 1
836 7 38 22
837 9 836 1
838 8 419 2
839 9 840 1
840 7 42 20
841 8 29 2
842 8 840 2
843 9 841 2
844 7 422 2
845 9 169 5
846 8 844 2
847 8 968 121
848 8 424 2
849 9 847 2
850 9 425 2
851 9 972 121
852 8 426 2
853 9 964 111
854 9 852 2
855 9 857 2
856 8 214 4
857 8 968 111
858 8 39 22
859 9 857 2
860 8 43 20
861 9 287 3
862 8 864 2
863 9 864 1
864 7 36 24
865 9 864 1
866 7 888 22
867 9 289 3
868 8 434 2
869 8 1738 2
870 8 872 2
871 8 1742 2
872 7 218 4
873 9 871 2
874 8 437 2
875 9 876 1
876 7 438 2
877 8 888 11
878 7 439 2
879 8 880 1
880 6 40 22
881 7 1762 2
882 6 441 2
883 7 1766 2
884 6 442 2
885 8 883 2
886 6 888 2
887 7 888 1
888 5 222 4
889 7 888 1
890 6 888 2
891 8 81 11
892 6 446 2
893 8 892 1
894 7 892 2
895 8 896 1
896 6 224 4
897 8 896 1
898 7 896 2
899 8 888 11
900 6 30 2
901 8 900 1
902 7 900 2
903 9 43 21
904 7 226 4
905 9 883 22
906 8 884 22
907 9 896 11
908 8 454 2
909 9 887 22
910 7 888 22
911 9 800 111
912 8 38 24
913 8 1826 2
914 8 892 22
915 9 913 2
916 8 458 2
917 10 473 444
918 8 896 22
919 9 920 1
920 7 46 20
921 9 800 121
922 7 924 2
923 8 924 1
924 6 42 22
925 8 484 441
926 7 924 2
927 9 483 444
928 7 464 2
929 9 484 445
930 8 484 446
931 9 932 1
932 7 466 2
933 9 932 1
934 8 932 2
935 9 924 11
936 7 468 2
937 9 936 1
938 8 936 2
939 9 961 22
940 8 470 2
941 9 942 1
942 7 964 22
943 9 942 1
944 7 946 2
945 8 946 1
946 6 968 22
947 8 946 1
948 7 946 2
949 9 946 3
950 7 972 22
951 9 729 222
952 7 476 2
953 8 964 11
954 8 952 2
955 8 957 2
956 7 478 2
957 7 968 11
958 7 960 2
959 8 957 2
960 6 480 2
961 7 31 2
962 6 964 2
963 7 964 1
964 5 482 2
965 7 964 1
966 5 968 2
967 6 968 1
968 4 484 2
969 6 968 1
970 5 968 2
971 7 968 3
972 5 486 2
973 7 972 1
974 6 972 2
975 8 964 11
976 6 488 2
977 8 966 11
978 7 976 2
979 7 968 11
980 7 490 2
981 8 968 13
982 8 960 22
983 8 972 11
984 7 492 2
985 9 963 22
986 7 964 22
987 9 964 23
988 7 966 22
989 8 967 22
990 6 968 22
991 8 968 23
992 7 968 24
993 9 968 25
994 7 972 22
995 9 972 23
996 8 498 2
997 9 999 2
998 8 976 22
999 8 111 9
1000 7 10 3
1001 8 1023 22
1002 7 1024 22
1003 8 1025 22
1004 8 502 2
1005 9 1003 2
1006 8 1008 2
1007 9 1008 1
1008 7 42 24
1009 9 888 121
1010 7 1012 2
1011 8 1012 1
1012 6 46 22
1013 7 2026 2
1014 7 1012 2
1015 8 1013 2
1016 7 254 4
1017 9 113 9
1018 8 1016 2
1019 8 1021 2
1020 7 510 2
1021 7 1023 2
1022 6 1024 2
1023 6 2046 2
1024 5 10 2
1025 6 2050 2
1026 6 1024 2
1027 7 1025 2
1028 7 514 2
1029 8 1025 4
1030 8 1024 6
1031 9 1023 8
1032 7 258 4
1033 8 1035 2
1034 8 47 22
1035 7 2070 2
1036 8 518 2
1037 8 1035 2
1038 9 554 484
1039 9 1023 16
1040 8 40 26
1041 9 1025 16
1042 9 521 2
1043 9 1021 22
1044 8 1022 22
1045 8 1023 22
1046 7 1024 22
1047 8 1025 22
1048 8 262 4
1049 9 968 81
1050 8 525 2
1051 9 1052 1
1052 7 526 2
1053 9 81 13
1054 7 527 2
1055 8 1056 1
1056 6 44 24
1057 7 2114 2
1058 6 529 2
1059 7 2118 2
1060 7 530 2
1061 8 1059 2
1062 7 531 2
1063 9 1059 4
1064 8 266 4
1065 9 1067 2
1066 8 533 2
1067 8 1089 22
1068 8 534 2
1069 8 2138 2
1070 9 535 2
1071 9 119 9
1072 9 134 8
1073 9 1089 16
1074 8 1296 222
1075 9 964 111
1076 9 538 2
1077 9 966 111
1078 8 49 22
1079 8 968 111
1080 8 45 24
1081 9 968 113
1082 9 576 506
1083 9 361 3
1084 9 1062 22
1085 8 1087 2
1086 9 1087 1
1087 7 1089 2
1088 8 1089 1
1089 6 33 2
1090 8 1089 1
1091 7 1089 2
1092 8 42 26
1093 8 1089 4
1094 9 1092 2
1095 8 1111 16
1096 8 548 2
1097 9 1089 8
1098 8 1100 2
1099 8 2198 2
1100 7 50 22
1101 8 2202 2
1102 8 551 2
1103 8 1111 8
1104 7 46 24
1105 8 1107 2
1106 8 1104 2
1107 7 1109 2
1108 7 554 2
1109 6 1111 2
1110 6 2220 2
1111 5 2222 2
1112 6 2224 2
1113 6 1111 2
1114 7 1112 2
1115 7 1111 4
1116 8 1112 4
1117 8 1111 6
1118 9 43 26
1119 8 1111 8
1120 8 224 5
1121 8 2242 2
1122 7 2244 2
1123 8 2246 2
1124 8 1122 2
1125 9 1109 16
1126 8 1148 22
1127 8 1111 16
1128 8 1130 2
1129 9 1107 22
1130 7 1152 22
1131 8 1109 22
1132 8 1110 22
1133 7 1111 22
1134 8 1112 22
1135 8 1111 24
1136 8 568 2
1137 9 968 169
1138 9 1136 2
1139 9 1141 2
1140 8 570 2
1141 8 1152 11
1142 8 1144 2
1143 9 1111 32
1144 7 44 26
1145 9 1024 121
1146 7 1148 2
1147 8 1148 1
1148 6 574 2
1149 8 1148 1
1150 6 1152 2
1151 7 1152 1
1152 5 576 2
1153 7 1152 1
1154 6 1152 2
1155 8 1111 44
1156 6 34 2
1157 8 1156 1
1158 7 1156 2
1159 9 1111 48
1160 7 580 2
1161 9 1150 11
1162 8 1160 2
1163 8 1152 11
1164 8 582 2
1165 9 1152 13
1166 9 53 22
1167 9 1156 11
1168 8 584 2
1169 9 2338 2
1170 8 1148 22
1171 10 1148 23
1172 8 1150 22
1173 9 1151 22
1174 7 1152 22
1175 9 1111 64
1176 8 196 6
1177 9 107 11
1178 8 1156 22
1179 10 957 222
1180 9 590 2
1181 10 1225 44
1182 9 960 222
1183 10 169 7
1184 8 592 2
1185 9 1296 111
1186 8 964 222
1187 10 964 223
1188 8 54 22
1189 9 967 222
1190 7 968 222
1191 9 968 223
1192 8 596 2
1193 10 968 225
1194 8 398 3
1195 9 1196 1
1196 7 598 2
1197 9 399 3
1198 8 1196 2
1199 8 109 11
1200 7 400 3
1201 9 1199 2
1202 8 1200 2
1203 9 401 3
1204 9 86 14
1205 9 241 5
1206 8 402 3
1207 10 964 243
1208 9 302 4
1209 9 968 241
1210 8 55 22
1211 9 968 243
1212 9 202 6
1213 9 729 484
1214 9 607 2
1215 9 243 5
1216 9 38 32
1217 9 1219 2
1218 9 1196 22
1219 8 1221 2
1220 8 2440 2
1221 7 111 11
1222 8 2444 2
1223 8 1221 2
1224 8 968 256
1225 7 35 2
1226 9 968 258
1227 8 1225 2
1228 9 614 2
1229 9 1225 4
1230 9 123 10
1231 9 2462 2
1232 8 44 28
1233 9 2466 2
1234 9 1012 222
1235 10 1013 222
1236 9 206 6
1237 10 1221 16
1238 9 1240 2
1239 9 2478 2
1240 8 62 20
1241 9 1243 2
1242 9 621 2
1243 8 113 11
1244 8 622 2
1245 9 1023 222
1246 8 623 2
1247 9 1025 222
1248 8 48 26
1249 8 2498 2
1250 7 625 2
1251 8 2502 2
1252 8 1250 2
1253 9 1251 2
1254 8 627 2
1255 10 968 287
1256 9 628 2
1257 10 419 3
1258 9 629 2
1259 10 1148 111
1260 9 42 30
1261 9 2522 2
1262 10 631 2
1263 9 1152 111
1264 9 79 16
1265 9 115 11
1266 8 2532 2
1267 9 2534 2
1268 9 784 484
1269 10 1148 121
1270 9 254 5
1271 10 1150 121
1272 8 1274 2
1273 9 1152 121
1274 7 1296 22
1275 9 1274 1
1276 8 58 22
1277 10 1156 121
1278 8 1280 2
1279 9 1280 1
1280 7 64 20
1281 9 1280 1
1282 8 1280 2
1283 9 1285 2
1284 8 800 484
1285 8 1296 11
1286 9 800 486
1287 9 117 11
1288 8 322 4
1289 10 1285 4
1290 8 1292 2
1291 9 1292 1
1292 7 1294 2
1293 8 1294 1
1294 6 1296 2
1295 7 1296 1
1296 5 6 4
1297 7 1296 1
1298 6 1296 2
1299 8 1296 3
1300 7 1296 4
1301 9 1296 5
1302 8 1296 6
1303 10 1292 11
1304 8 326 4
1305 9 1294 11
1306 9 653 2
1307 8 1296 11
1308 8 218 6
1309 8 119 11
1310 8 1332 22
1311 9 1089 222
1312 8 1296 16
1313 10 101 13
1314 9 219 6
1315 9 1331 16
1316 8 1294 22
1317 9 439 3
1318 7 1296 22
1319 9 1296 23
1320 7 220 6
1321 9 1320 1
1322 8 1320 2
1323 8 441 3
1324 9 662 2
1325 9 1323 2
1326 8 221 6
1327 8 1329 2
1328 8 664 2
1329 7 1331 2
1330 7 1332 2
1331 6 11 3
1332 6 222 6
1333 7 1331 2
1334 7 1332 2
1335 8 1331 4
1336 8 668 2
1337 9 1331 6
1338 8 223 6
1339 9 1331 8
1340 8 1296 44
1341 9 1352 11
1342 8 1344 2
1343 9 1332 11
1344 7 224 6
1345 9 1344 1
1346 8 1344 2
1347 9 1331 16
1348 7 674 2
1349 9 1348 1
1350 7 1352 2
1351 8 1352 1
1352 6 676 2
1353 8 123 11
1354 7 1352 2
1355 9 1331 24
1356 7 678 2
1357 9 1356 1
1358 8 1356 2
1359 10 1348 11
1360 8 680 2
1361 10 1350 11
1362 8 1364 2
1363 9 1352 11
1364 7 62 22
1365 9 1364 1
1366 8 1364 2
1367 8 1369 2
1368 8 968 400
1369 7 37 2
1370 9 886 484
1371 8 1369 2
1372 8 888 484
1373 9 1369 4
1374 8 1152 222
1375 9 125 11
1376 8 86 16
1377 10 81 17
1378 8 1600 222
1379 10 1331 48
1380 9 46 30
1381 11 896 485
1382 10 896 486
1383 10 461 3
1384 9 346 4
1385 10 1369 16
1386 8 63 22
1387 10 1386 1
1388 9 1386 2
1389 9 22224 16
1390 9 1412 22
1391 9 1369 22
1392 9 58 24
1393 10 1152 241
1394 9 697 2
1395 10 1152 243
1396 8 698 2
1397 9 127 11
1398 9 233 6
1399 9 2798 2
1400 8 350 4
1401 10 1399 2
1402 9 1400 2
1403 10 1404 1
1404 8 702 2
1405 9 1406 1
1406 7 1408 2
1407 8 1408 1
1408 6 64 22
1409 8 1408 1
1410 7 1408 2
1411 9 1408 3
1412 7 706 2
1413 9 1412 1
1414 8 1412 2
1415 10 1294 121
1416 8 354 4
1417 9 109 13
1418 9 1416 2
1419 9 129 11
1420 9 710 2
1421 10 1408 13
1422 8 1444 22
1423 10 1023 400
1424 9 89 16
1425 10 475 3
1426 9 62 23
1427 10 1449 22
1428 9 42 34
1429 10 1407 22
1430 8 65 22
1431 10 1408 23
1432 9 1408 24
1433 9 1444 11
1434 9 239 6
1435 10 287 5
1436 9 359 4
1437 10 479 3
1438 9 1440 2
1439 10 1440 1
1440 8 40 36
1441 9 1442 1
1442 7 1444 2
1443 8 111 13
1444 6 38 2
1445 8 1444 1
1446 7 482 3
1447 9 1444 3
1448 8 964 484
1449 8 483 3
1450 7 1452 2
1451 8 1452 1
1452 6 484 3
1453 8 1452 1
1454 7 1452 2
1455 8 485 3
1456 8 972 484
1457 9 1455 2
1458 7 486 3
1459 9 1458 1
1460 8 1458 2
1461 9 487 3
1462 8 731 2
1463 9 133 11
1464 8 244 6
1465 10 1024 441
1466 8 1444 22
1467 10 489 3
1468 9 734 2
1469 9 113 13
1470 9 245 6
1471 10 1449 22
1472 8 46 32
1473 10 1352 121
1474 8 1452 22
1475 10 1452 23
1476 9 123 12
1477 10 1455 22
1478 9 1480 2
1479 9 1600 121
1480 8 740 2
1481 10 1479 2
1482 9 1480 2
1483 10 1485 2
1484 9 742 2
1485 9 495 3
1486 9 1488 2
1487 10 1485 2
1488 8 62 24
1489 9 1600 111
1490 9 1488 2
1491 10 497 3
1492 8 746 2
1493 10 964 529
1494 9 1492 2
1495 9 1936 441
1496 8 44 34
1497 9 968 529
1498 9 1496 2
1499 9 1521 22
1500 9 500 3
.... ... ... ...
Based on it, we easily can reconstruct decomposition of each number of the table:
$567 = 565+2 = 576 - 11+2 = 24^2 - 22/2+2 = (22+2)^2-22/2+2$ $\;$: requires $8$ $\;2$s.
Note that "minimal" decompositions of some numbers $n$ require essentially large (in comparison with $n$) parts:
$101 = 2222/22$;
$825 = 1936 - 1111 = (2\times 22)^2 - 2222/2$.
On upper bound.
If for some $n_0$ for all $n \in [n_0, n_0^2]$ works estimation $$ g(n) \le c \log_2 n - 4, \tag{1} $$ then it works for all $n \ge n_0$.
Indeed,
for any $n\in [n_0^2, n_0^4]$ we can express $n$ as:
$$
n = a^2+b,
$$
where
$a = \lfloor \sqrt{n} \rfloor$,
$b = n-a^2 \le 2a\;\;$ (the worst case: when $n=(a+1)^2-1$; then $n-a^2=2a$).
Then in the case of even $b$: $b=2s$: $$ g(b) = g(2s) = 1+g(s) \le 1+g(a); $$ and in the case of odd $b$: $b=2s+1$: $$ g(b)=g(2s+2/2) = 3+g(s) \le 3+g(a); $$ and therefore $$ g(n) = g(a^2+b) \le 1 + g(a) + g(b) \le 1 + g(a) + 3+ g(a) = 4+2g(a), $$ so since $a\in [n_0, n_0^2]$, we rewrite it: $$ g(n) \le 4+2(c \log_2 a - 4) = c \log_2 a^2 - 4 \le c \log_2 n - 4. $$ Based on math.induction, we can expand it for any $n\ge n_0$.
It remains to figure out appropriate value $c$.
Experimentally (see previous answer) for all $n\in [400, 400^2]$ works estimation: $$ g(n) \le 1.5 \log_2 n - 4;\tag{2} $$ therefore for all $n\ge 400$ we can use rough estimation $(2)$.
Check for $n=14885392687$: $g(n) < 46.7$ (not so big improvement ...).
Similarly, the estimation $$ g(n) \le 1.2 \log_2 n - 4;\tag{2'} $$ works for $n\ge 20\;000$.
Check for $n=14885392687$: $g(n) < 36.6$ (slightly better improvement).
I am pretty sure that starting from some $n_0$ we can use value $c=1$, or even less (?)