AMC 12 2009a 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

$$\underbrace{\log_2\log_2\log_2\ldots\log_2B}_{k\text{ times}}$$

is defined?

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