It seems like most natural situations (that at least I've come across) where Euler's number shows up, there's usually a more general (yet still natural) version of it that contains the exponential function. Even its usual definition as: $$e = \lim_n\left(1+\frac{1}{n}\right)^n$$ Can be extended to: $$e^x = \lim_n\left(1+\frac{x}{n}\right)^n$$
This seems, at least to me, somewhat unique. Rarely do I find results with other constants like $\pi$ where I can naturally extend them to a consistent family of values like how I can with $e \to e^x$.
Thus the question, does Euler's number have any significance that can't be naturally extended to other values of the exponential? Or alternatively, is there any significance to Euler's number other than being equal to $\exp(1)$ or the base of the exponential function?
Note: I realize of course that this isn't really a mathematical question, and that this might not be the appropriate place for this sort of question, if that's indeed the case I will remove this post.