2009 AMC 12A Problem 24

The ‘‘tower function of twos’’ is defined recursively as follows: $T(1) = 2$ and $T(n + 1) = 2^{T(n)}$ for $n\ge1$. Let $A = (T(2009))^{T(2009)}$ and $B = (T(2009))^A$. What is the largest integer $k$ such that

$\log_2\log_2\log_2\ldots\log_2B_{k}$where there are $k$ $\log_2$'s, is defined?

$\textbf{(A)}\ 2009\qquad \textbf{(B)}\ 2010\qquad \textbf{(C)}\ 2011\qquad \textbf{(D)}\ 2012\qquad \textbf{(E)}\ 2013$

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.