Insane Numbers

Let $x = 10^9!$ and $y = 10^{{10}^{10}}$ .

We compare $x$ and $y$ by taking logarithms (with base $10$ ).

By the laws of logarithm, we have $\log x = \log 10^9! = \sum_{n=1}^{10^9} \log n < \sum_{n=1}^{10^9} \log 10^{10} = \sum_{n=1}^{10^9} 10 = 10^9 \cdot 10 = 10^{10} = \log y,$ which implies that $x < y$ as $\log(\, \cdot \,)$ is an increasing function.

Hence, $y = \boxed{\mathbf{10^{10^{10}}} \textbf{ is bigger}}.$

1 000 000 000! < $1 000 000 000^{1 000 000 000}$ -> 9 zeros * 1 billion -> this number has lesser than 9 billion zeros

on the other side

$10^{10^{10}}$ = $10^{10 000 000 000}$ -> 1 zero * 10 billion -> this number has exactly 10 billion zeros