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

$$e^x:=\sum_{k=0}^{\infty}\frac{x^k}{k!}$$
also
$$e^x:=\lim_{n\to\infty}\bigg(1+\frac{x}{n}\bigg)^n$$
or this one:
Define $e^x:\mathbb{R}\rightarrow\mathbb{R}\\$ as unique function satisfying:
\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).
The exponential function is the unique solution of the initial value problem

$y'(x)=y(x) , \quad y(0)=1$.
Here's an "unusual" one: $e$ is the positive real number such that

$$
\sqrt{6\log\left(e\sqrt[4]{e}\sqrt[9]{e}\sqrt[16]{e}\sqrt[25]{e}\ldots\right)} = \pi.
$$