Why is Euler's number $2.71828$ and not anything else? Why is Euler's number $\mathtt 2.71828$ and not for example $\mathtt 3.7589$?

I know that $e$ is the base of natural logarithms. I know about areas on hyperbola xy=1 and I know its formula: $$e =\sum_{n=0}^\infty \frac{1}{n!} \approx 2.71828$$
And I also know it has many other characterizations.
But, why is $e$ equal to that formula (which sum is approximately $\mathtt 2.71828$)?

I googled that many times and every time it ends in having "$e$ is the base of natural logarithms". I don't want to work out any equations using $e$ without understanding it perfectly.

Summary: I'm looking for the origin of $e$, if $\pi$ came from the radius of a circle with a unit diameter, then what is $e$ ???
$\sum\frac1{n!}$ is not that special.

$\lim_{n\to\infty}\left(1+\frac1n\right)^n$ is not really special.

$f'(x)=f(x)$ is a very simple differential equation, but unremarkable, really.

$\ln (x)$ is only marginally nicer than other logarithms, in that its derivative is $\frac1x$.

The fact that a single number connects all of these (and many, many others) as intimately as $e$ does is nothing short of a miracle. Oh, and also $e$ happens to have the decimal expansion $2.718\ldots$
One introduction of $e$ can be seen as a solution to $f'(x)=f(x)$. Indeed with $\lambda$ any constant, the function
$$ f \ : \ x \mapsto \lambda e^x$$ 
is so that at each point, its "increase" equals its value, hence $f'(x)=f(x)$