2009 AMC 12A Question 24

The tower function of twos is defined recursively as follows: $T(1) = 2$ and $T(n + 1) = 2^{T(n)}$ for $n \ge 1.$ Let $A = (T(2009))^{T(2009)}$ and $B = (T(2009))^A.$ What is the largest integer $k$ such that $$
\underbrace{\log_2 \log_2 \log_2 \ldots \log_2}_{k} B
$$ is defined?