# Possible definitions of exponential function

I was wondering how many definitions of exponential functions can we think of. The basic ones could be:

$$ex:=∞∑k=0xkk!e^x:=\sum_{k=0}^{\infty}\frac{x^k}{k!}$$
also
$$ex:=limn→∞(1+xn)ne^x:=\lim_{n\to\infty}\bigg(1+\frac{x}{n}\bigg)^n$$
or this one:
Define $$ex:R→Re^x:\mathbb{R}\rightarrow\mathbb{R}\\$$ as unique function satisfying:
ex≥x+1∀x,y∈R:ex+y=exey\begin{align} e^x\geq x+1\\ \forall x,y\in\mathbb{R}:e^{x+y}=e^xe^y \end{align}
Can anyone come up with something unusual? (Possibly with some explanation or references).

$$y′(x)=y(x),y(0)=1y'(x)=y(x) , \quad y(0)=1$$.