I haven't studied number theory, but I expect that someone who has would be able to answer this. Successive powers of three have final digits in the repeating pattern 1, 3, 9, 7, so if we can find N mod 4 for the N such that 3^N = 3^^^3, then we would have our answer.
3^odd = 3 mod 4
so it ends in 7.
(but I repeat myself)
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