HMMT November 2012 H3

Find the smallest positive integer $n$ such that $\underbrace{2^{2 \cdot 2}}_{n}>3^{3^{3^{3}}}$. (The notation $\underbrace{2^{2 \cdot 2^{2}}}_{n}$ is used to denote a power tower with $n 2$ 's. For example, $\underbrace{2^{2^{2}}}_{n}$ with $n=4$ would equal $2^{2^{2^{2}}}$.)