\uline{The tower function of twos} is defined recursively as follows: $$T\left(1\right)=2$$ and $$T\left(n+1\right)=2^{T\left(n\right)}$$ for $$n\geqslant 1$$. Let $$A=\left(T\left(2009\right)\right)^{T\left(2009\right)}$$ and $$B=\left(T\left(2009\right)\right)^{A}$$. What is the largest integer $$k$$ such that $$\underbrace{\log_{2}\log_{2}\log_{2}\ldots\log_{2}B}_{{k\textasciitilde{} \text{times}}}$$ is defined?  2的\uline{塔函数}定义如下: $$T\left(1\right)=2$$ , $$T\left(n+1\right)=2^{T\left(n\right)}$$ ($$n\geqslant 1$$). 令 $$A=\left(T\left(2009\right)\right)^{T\left(2009\right)}$$ , $$B=\left(T\left(2009\right)\right)^{A}$$. 使得 $$\underbrace{\log_{2}\log_{2}\log_{2}\ldots\log_{2}B}_{{k\textasciitilde{} \text{times}}}$$有意义的最大整数 $$k$$是多少?
选项为:A:$$2009$$  B:$$2010$$  C:$$2011$$  D:$$2012$$  E:$$2013$$  请回答其中一个选项.