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.



Repeated logarithm of a power tower
The answer is "no" ... also see a Usenet sci.math discussion in July, 2009.


