Which is larger, $A=20^{19^{20}}$ or $B=19^{20^{19}}$ ? (Here, $a^{b^{c}}$ means $a^{\left(b^{c}\right)}$ and not $\left(a^{b}\right)^{c}$.)
The answer is that $A$ is larger. First,

$$
\begin{aligned}
\left(\frac{19}{20}\right)^{20} & =0.95^{20}=0.9025^{10}>0.9^{10} \\
& =0.81^{5}=0.8^{5}=0.32768>0.05=\frac{1}{20}
\end{aligned}
$$

which implies that $19^{20}>20^{19}$. Thus, $A=20^{19^{20}}>20^{20^{19}}>19^{20^{19}}=B$, as desired.