Question 102
Find the smallest positive integer $n$ such that
$$
\underbrace{2^{2^{2} \cdots^{2}}}_{n\ 2\text{'s}} \gt \underbrace{((\cdots((100!)!)!\cdots)!)!}_{100 \text{ factorials }}
$$
$$
\underbrace{2^{2^{2} \cdots^{2}}}_{n\ 2\text{'s}} \gt \underbrace{((\cdots((100!)!)!\cdots)!)!}_{100 \text{ factorials }}
$$