$e$ has many definitions and properties. The one I'm most used to is

$$\lim_{n\to \infty}\left(1+\frac{1}{n}\right)^n $$

If someone asked me (and I didn't know about $e$):

Is there a constant $c$ such that the equation $\frac{d}{dx}c^x=c^x $ is true for all $x$?

Then I'd likely answer that:

I doubt it! That would be a crazy coincidence.

I'm curious, is it a coincidence that there is a constant that makes this true?