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 $A=T(2000)$ and $B=(T(2000){)}^A$. What is the largest integer $k$ such that

$\underset{ktimes}{\underbrace{{\log }_2{\log }_2{\log }_2\cdots {\log }_{2}B}}$

is defined?