Any closed form known for $\lim \limits_{n \to \infty}\ \underbrace{\log_2 \log_2 \dots \log_2}_n\ \underbrace{3^{3^{\cdot^{\cdot^{3}}}}}_n$? Numerical evidence suggests that it is around $2.4440214614892$ and converges very fast, but I was not able to prove existence of this limit.
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1$\begingroup$ For what it's worth, it doesn't show up in Steven Finch's book, Mathematical Constants, and the sequence of digits doesn't show up in the Online Encyclopedia of Integer Sequences. Plouffe's Inverter also came up empty. $\endgroup$– Gerry MyersonCommented Feb 8, 2011 at 22:41
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2 Answers
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The answer is "no" ... also see a Usenet sci.math discussion in July, 2009.
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$\begingroup$ I think the discussion GE refers to is the one with subject header "Logarithm of repeated exponential." mathforum.org/kb/thread.jspa?messageID=6786389&tstart=0 is one way to get there. $\endgroup$ Commented Feb 9, 2011 at 4:32
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Canceling the innermost operations, you see that the sequence is increasing. On the other hand, the related sequence in which you replace the tower of $3$'s by $10$ times that tower is decreasing for large $n$ (you can again cancel the innermost operations, but now with loss). This justifies both convergence and high speed (indeed, the factor of $10$ next to the tower of $3$'s is not going to be very noticeable after all those logarithms).
I doubt very much you have a nice closed form answer here and the whole thing looks more appropriate for AoPS than for MO, but OK, here are my 2 cents :).