By the way, here's the handwavy heuristic version that gives the right answer.

When you are comparing two exponentials with really big exponents, only the exponent really matters even if the difference in bases is huge (unless it's really huge or the exponents are really close). So the following should all be the same:

• 3^(n)^3 versus G^10^10^100 where G = 10^10^100
• 3^(n-1)^3 versus 10^10^100
• 3^(n-2)^3 versus 10^100
• 3^(n-3)^3 versus 100
  • NB you can do all the foregoing in a single step: just cross out the 3 bottom levels.

and now we can see that 3^3 < 100 but 3^3^3 > 100, so n-3=3 and n=6.

If the detailed algebra in the parent isn't enough to make "look at the exponent and ignore the bases" plausible, here's another way to see it that happens to work neatly in this case:

(10^10^100) ^ (10^10^100) = 10^(10^100 . 10^10^100) = 10^10^(100 . 10^100) = 10^10^10^102

so the difference between 10 and 10^10^100 on the base is the same as the difference between 100 and 102 three levels up!

... EDIT, on happening to reread this days later: No, I slipped up in the calculation above and the result is actually a lot closer.

(10^10^100) ^ (10^10^100) = 10^(10^100 . 10^10^100) = 10^10^(100 + 10^100) = 10^10^10^(100+teenytiny)

because 10^100 + 100 is barely bigger than 10^100 at all. In fact it turns out that "teenytiny" is about 4 x 10^-99. So: the difference between 10 and 10^10^100 on the base is the same as the difference between 100 and 100 + 4e-99 three levels up.