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

\underline{\log_2\log_2\log_2\ldots\log_2B}_{\,k\text{ times}}

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.