Q:

How to prove which of two numbers written as powers is bigger? 

Prove which number is larger:
a) $10^{100!}$ or $10^{10^{100}}$
b) $e^\pi$ or $\pi^e$
I know we all know how to plug these into the calculator and check, but how someone mathematically prove which one is bigger with words and calculations?  

A:

For (a), it boils to comparing $1$ and
$$
\frac{100!}{10^{100}}=\frac{1}{10}\cdot\frac{2}{10}\cdot\cdots\frac{100}{10}.\tag{*}
$$
Only the fractions $\frac{1}{10},\ldots,\frac{9}{10}$ are less than $1$ but they are more than compensated for by $\frac{100}{10},\frac{90}{10},\ldots,\frac{20}{10}$. Other fractions are greater or equal to $1$. So it should be clear that the expression in (*) is greater than $1$. This of course means $10^{100!}>10^{10^{100}}$.
For (b), an answer is already linked to in the comment or you can look at Edward Jiang's answer.