Generate Programs in Increasing size

CJam, 10 bytes

{"_~"+_}_~

Test it here.

Explanation

{"_~" e# Generalised quine framework, leaves the block and the string "_~"
      e# on the stack. 
+     e# Prepend the block to the string.
_     e# Duplicate the resulting array.
}_~

JavaScript, 62 61 37 bytes

Thanks to @Doᴡɴɢᴏᴀᴛ for the help!

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Original [37 bytes]:

f=_=>'f='+'_'.repeat((10+f).length)+f

Child [74 bytes]:

f=______________________________________=>'f='+'_'.repeat((10+f).length)+f

Grandchild [148 bytes]:

f=________________________________________________________________________________________________________________=>'f='+'_'.repeat((10+f).length)+f

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Alternate (with printing to console, and as a full program):

Original [61 bytes]:

f=_=>console.log(`f=${'_'.repeat((0+f).length+5)+f};f()`);f()

Child [122 bytes]:

f=______________________________________________________________=>console.log(`f=${'_'.repeat((0+f).length+5)+f};f()`);f()

Grandchild [244 bytes]:

f=________________________________________________________________________________________________________________________________________________________________________________________=>console.log(`f=${'_'.repeat((0+f).length+5)+f};f()`);f()

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How it works!

1.   f=_=>   Define function f as console.log(...)

2.   ;f()   Run function f.

3.   (in function f)  

• console.log(...)   Print the following:

  • f=   literal text "f="
  • ${'_'.repeat((0+f).length+5)   "_" repeated for the length of f, altered to account for characters not included in the stringification of f
  • +f}   The stringification of function f
  • ;f()   literal text ";f()"

Notes

• console.log is necessary instead of alert because alert doesn't seem to play well with really long strings (at least on my machine/browser configuration)
• The _'s are inserted into the name of the (unused) parameter of function f, to ensure that they are included in the stringification of f.
• Main improvement (aside from getting rid of the console.log) of the first solution over the second: adding 10 to the function instead of 0 to cast it to string makes it one byte longer, eliminating the need to add 1 to the length afterwards, saving a byte.

Minkolang 0.15, 19 14 bytes

"66*2-rIDdr$O.

Original, child, grandchild.

Explanation

"66*2-      $O.    Standard quine formulation

      r            Reverse stack
       I           Push length of stack
        D          Pop n and duplicate top of stack n times
         d         Duplicate top of stack
          r        Reverse stack

What the bit in between rs does is duplicate the ending period enough times to fulfill the doubling criterion. . is the "stop program" character, so the many periods at the end do nothing except be there.