Print this maximally long URL

Python 3, ^{86 \$\cdots\$ 81} 77 bytes

Saved 3 bytes thanks to mypetlion!!!
Saved 4 bytes thanks to Shieru Asakoto!!!

print(f"http://{'.'.join(chr(i+97)*int(1+.7**-i/120)for i in range(26))}.me")

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Uses Shieru Asakoto's 0-based formula.

APL (Dyalog), 41 38 bytes

-3 bytes thanks to Bubbler

2⌽'mehttp://',∊'.',¨⍨⎕A⍴¨⍨⌊1+62×.7*⍒⎕A

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Outputs the URL with the letters capitalised. Uses the 0 indexed formula \$ \lfloor 1 + 62 \times 0.7^{25-x} \rfloor \$, since \$ (\frac{10}{7})^{x-25} = ((\frac{7}{10})^{-1})^{x-25} = (\frac{7}{10})^{25-x}\$

Explanation:

2⌽                         ⍝ Rotate by two places (moving the 'me' to the end)
  'mehttp://'              ⍝ The start string
             ,             ⍝ Followed by
              ∊               ⍝ The flattened string of
               '.'            ⍝ A period
                  ,⍨¨         ⍝ Appended to the end of each of
                     ⎕A            ⍝ The uppercase alphabet
                       ⍴¨⍨         ⍝ Where each letter is repeated by
                          ⌊               ⍝ The floor of
                           1+             ⍝ 1 plus
                             62×          ⍝ 62 times
                                .7*       ⍝ 0.7 to the power of
                                   ⍒⎕A  ⍝ The range 26 to 1

05AB1E, 25 24 23 21 bytes

žXAS.7Ƶ-t∞-m>×….me¬ýJ

-1 byte thanks to @Neil's analysis that *(10/7)** is the same as /.7**.
-3 bytes thanks to @Grimmy using a different formula and ingenious use of ý!

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Explanation:

The formula used to get the correct amount of characters of the alphabet, where \$n\$ is the 1-based index of the alphabetic letter:
\$a(n) = \left\lfloor0.7^{(\sqrt{208}-n)}+1\right\rfloor\$

žX                     # Push builtin "http://"
      Ƶ-               # Push compressed 208
        t              # Take the square-root of that: 14.422...
         ∞             # Push an infinite list of positive integers: [1,2,3,...]
          -            # Subtract each from the 14.422...: [13.442...,12.442...,...]
    .7     m           # Take 0.7 to the power of each of these: [0.008...,0.011...,...]
            >          # Increase each by 1: [1.008...,1.011...,...]
  AS                   # Push the lowercase alphabet as list of characters,
             ×         # and repeat each the calculated values amount of times as string
                       # (which implicitly truncates the floats to integers, and ignores
                       #  the floats beyond the length of the alphabet)
              ….me     # Push ".me"
                  ¬    # Push its head (the "."), without popping the ".me" itself
                   ý   # Join with delimiter. Normally it will use the top value as
                       # delimiter and joins the top-1'th list. In this case however, the
                       # top-1'th item is a string, so instead it will join the entire stack
                       # together. BUT, because the stack contains a list, it will instead
                       # only join all lists on the stacks by the "." delimiter
                    J  # And finally join the three strings on the stack together
                       # (after which this result is output implicitly)

See this 05AB1E tip of mine (section How to compress large integers?) to understand why Ƶ- is 208.