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Manfred comments on A math question - Less Wrong
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Yes. 3^^^4 < 4^^^4, and the denominator has the correct number and type of operations to be 3^^^4, since 3^^^4 = 3^^3^^3^^3 - however it's actually smaller. Up arrow notation evaluates from right to left, which results in the largest possible number. The parentheses just muck things up and make the result smaller - so the answer is much much greater than 1.
So 3^^^4 is an exponential tower of 3s 3^^4 levels tall?
Hm, let's find out. (Open this if you want to have a go at finding that answer first)
3^^X is an exponential tower of 3s that is X high. 3^^^4 = 3^^3^^3^^3 (i.e. it's 3 ^^ itself, with four 3s, just like 3^^4 is 3 ^ itself with four 3s). So 3^^^4 is an exponential tower of 3s that is 3^^3^^3 high. 3^^3^^3 is 3 ^^ itself 3 times, so it's 3^^^3.
So 3^^^4 is an exponential tower of 3s 3^^^3 levels tall.
Thanks for the explanation. It was worth the brain explosion.