Very Large Number sequences

[Ed Pegg]

  I've wondered about the properties of Very Large Numbers,
as described by Conway/Guy on page 61 of The Book of Numbers.
5->5->5->5 is an incomprehensibly huge number, but it seems
feasible to compute the sequence 5->5->5->5 (mod n) for n
from 1 to 30.

  Or perhaps 4 (mod 3), 4->4 (mod 3), 4->4->4 (mod 3),
4->4->4->4 (mod 3), 4->4->4->4->4 (mod 3), etc.

  For a Google, this is easy.
Table[PowerMod[10, 100, n], {n, 2, 50}]
{0, 1, 0, 0, 4, 4, 0, 1, 0, 1, 4, 3, 4, 10, 0, 4, 10, 9, 0, 4, 12, 13, 16,
0,
16, 10, 4, 16, 10, 5, 0, 1, 4, 25, 28, 10, 28, 16, 0, 1, 4, 31, 12, 10, 36,
27, 16, 11, 0}
A066298 -- already in OEIS

  For a Googleplex, this is easy.
Table[PowerMod[10, 10^100, n], {n, 2, 50}]
{0, 1, 0, 0, 4, 4, 0, 1, 0, 1, 4, 3, 4, 10, 0, 1, 10, 9, 0, 4, 12, 13, 16,
0,
16, 10, 4, 24, 10, 5, 0, 1, 18, 25, 28, 10, 28, 16, 0, 1, 4, 24, 12, 10, 36,
9, 16, 4, 0}

  So the moduli of a Google and a Googleplex differ for
17, 29, 34, 43, 47, 49, 51, 53, 58, 59, 68, 71, 73, 79, 83 ...
  --Ed Pegg Jr.