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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The answer is "no" ... also see a Usenet sci.math discussion in July, 2009. 


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 :). 

