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 :).