The tower function of threes is defined recursively as follows: $T(1)=3$ and $T(n+1)=3^{T(n)}$ for $n\ge 1$. Let $A=(T(2020){)}^{T(2020)}$ and $B=(T(2020){)}^A$. What is the largest integer $k$ such that

$\underset{ktimes}{\underbrace{{\log }_3{\log }_3{\log }_3\cdots {\log }_{3}B}}$

is defined?