Thetower function of twosis defined recursively as follows: $\displaystyle T(1) = 2$ and $\displaystyle T(n+1) = 2^{T(n)}$ for $\displaystyle n\geq1$. Let $\displaystyle A = (T(2009))^{T(2009)}$ and $\displaystyle B = (T(2009))^A$. What is the largest integer $\displaystyle k$ such that

$\displaystyle \underbrace{\log_{2}\log_{2}\log_{2}...\log_{2}} B $

$\displaystyle k$ times

is defined?