Denote the number of arrangements of $n$ envelopes as $K_n$.
Assume we are given an arrangement of $n$ envelopes. If we remove the smallest envelope, we obtain a possible arrangement of $n-1$ remaining envelopes. On the other hand, if we are given a possible arrangement of the remaining $n-1$ envelopes, we can put the smallest one directly in the biggest one, or in any of the remaining $n-2$ envelopes. Therefore, $K_n$ is exactly $n-1$ times larger than $K_{n-1}$.
We conclude that $K_n = (n-1) \cdot K_{n-1} = \cdots = (n-1) \cdot (n-2) \cdots 2 \cdot K_2$.
For $n=2$, the only possible arrangement is putting the smaller envelope inside the larger one, so $K_2=1$. Therefore $K_n = (n-1) \cdot (n-2) \cdots 1$.
For 100 envelopes, the number of different arrangements is
$$K_{100} = 99 \cdot 98 \cdots 1 = 99!$$