How many different definitions of $e$ are there?
You really should be looking for definitions of the exponential function $e^x$, not definitions of $e$. Here are the most important ones that come to mind:
- $e^x$ is the unique function $f(x)$ satisfying $f'(x) = f(x)$ and $f(0) = 1$.
- $e^x$ is the inverse of the function $\displaystyle \ln x = \int_1^x \frac{dt}{t}$.
- $e^x$ is the power series $\displaystyle \sum_{n \ge 0} \frac{x^n}{n!}$.
- $e^x$ is the limit $\displaystyle \lim_{n \to \infty} \left( 1 + \frac{x}{n} \right)^n$.
I say this because once you start thinking about $e^x$, which is by far the more fundamental object, and not $e$, the relationship between the definitions becomes much more transparent. Here are short sketches of proofs that the definitions above are all equivalent:
$1 \Leftrightarrow 2$: if $\frac{d}{dx} g(x) = \frac{1}{x}$ then
$$\frac{d}{dx} g^{-1}(x) = \frac{1}{g'(g^{-1}(x))} = g^{-1}(x).$$
Conversely, if $\frac{d}{dx} g(x) = g(x)$ then
$$\frac{d}{dx} g^{-1}(x) = \frac{1}{g'(g^{-1}(x))} = \frac{1}{g(g^{-1}(x))} = \frac{1}{x}.$$
$1 \Leftrightarrow 3$: if $f'(x) = f(x)$ then $f^{(n)}(x) = f(x)$ for all $n$, hence $f^{(n)}(0) = f(0) = 1$, so the Taylor series of $f(x)$ has all coefficients equal to $1$. Conversely, the function with that Taylor series is its own derivative and satisfies $f'(0) = 1$ using the fact that power series are term-by-term differentiable inside their interval of convergence.
$1 \Leftrightarrow 4$: $\displaystyle \lim_{n \to \infty} \left( 1 + \frac{x}{n} \right)^n$ is the result of using Euler's method to compute $f(x)$, where $f$ satisfies $f'(t) = f(t)$ and $f(0) = 1$, as the step size $n$ goes to $\infty$.
(My personal opinion is that the first definition is the most fundamental one; in general uniqueness statements are very powerful. For example, there is a very short proof using the first definition, which I invite you to find, that $e^{x + y} = e^x e^y$.)
$$e=\lim_{n\to\infty}\sqrt[\large^n]{\text{LCM}[1,2,3,\ldots,n]},$$ where LCM stands for least common multiple.
Actualy for certain numbers like $e$ and $\pi$ an almost enormous amount of defining relations or identities can be produced.
Think of the case of $\pi$, by using trigonometric identities (or even number-theoretic identities) many many defining relations for $\pi$ can be produced (i think there is also a systematic procedure to produce new defining relations).
A very close case is for $e$, due to being related to similar hyperbolic trigonometric functions (except the purely exponential-analytic identities), a very large amount of identities which can be used as definitions or representations can be found.
Some of them can be found online as in here, however they do not exhaust all possible identities that can serve as definitions.
Finally an identity that relates $i$, $\pi$, $e$, $0$ and $1$ (which was a favorite of R. Feynman) is the Euler identity:
$$e^{i\pi}+1=0$$