# Repeated logarithm of a power tower

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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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. –  Gerry Myerson Feb 8 '11 at 22:41

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).