I recently came across the following beautiful and seemingly out-of-the-blue appearance of $e$:
$E[\xi]=e$, where $\xi$ is a random variable that is defined as follows. It's the minimum number of $n$ such that $\sum_{i=1}^n r_i>1$ and $r_i$ are random numbers from uniform distribution on $[0,1]$.
I can think of several more almost magical applications of $e$,$^\dagger$ but I would like to hear of some instances where you were surprised that $e$ was involved.
I would like to collect the best examples so as to be able to give some of the high-school students I tutor in math a sense of some of the deep connections between different areas of math that only really become apparent at the university level. These deep connections have always made me want to learn more about math, and my hope is that my students would feel the same way.
EDIT (additional question): A lot of the answers below come from statistics and/or combinatorics. Why is $e$ so useful in these areas? In general, I'd very much appreciate if answerers included some pointers as to how one can get an intuition about why $e$ appears in their case (or indeed, how they themselves make sense of it) – this would greatly help me in presenting these great examples.
$^\dagger$For instance that its exponential function is its own derivative, its relation to the trigonometric functions, its use in Fourier transformation, transcendence, etc., all of which I must admit I don't really understand (perhaps except for the first one, which I take to be the definition of $e$), as in "what is it about $e$ that makes it perfect for representing complex numbers, or changing from one basis to another, etc.?"
Best Answer
We have
where LCM stands for least common multiple. The proof can be found here.