Why is $e$ so ubiquitous? $e$ is one of the most important numbers in our universe, it is everywhere. When I try to find out why the most common explanation is some reverse-engineered physics or finance problem. But these are just one off examples of why $e$ is important, they fall short of illuminating the origin of its significance. What I'm looking for is some fundamental definition of $e$ that explains it's significance and omnipresence, something akin to $\pi$ relating circumference and diameter. 
Thanks, I hope I'm being clear.
EDIT*: A common saying is "$e$ is the most natural base." What does that mean?
 A: $e^{\lambda x}$ is the solution to the equation $Df=\lambda f$. This is equivalent to saying that $e^{\lambda x}$ is an eigenfunction of the differentiation operator D for any value $\lambda \in \mathbb{R}$. Now how ubiquitous is differentiation to physics?
A: \begin{align}
\frac d {dx} 10^x & = ( 10^x\cdot\text{constant}) \approx 10^x \cdot (2.3) \\[10pt]
\frac d {dx} 2^x & =  ( 2^x \cdot \text{constant}) \approx 2^x\cdot (0.693)
\end{align}
etc.  It is easy to show that the derivative of an exponential function is a constant multiple of the same exponential function.
Only when the bas is $e$ is the "constant" equal to $1$.
The fact that $x\mapsto e^x$ is its own derivative accounts for its incessant appearance in the study of differential equations.  It also accounts for the fact that the "constant" is the base-$e$ logarithm of the base of the epxonential function.
That's the beginning of the story; there's a lot more to it.
The fact that the "constant" is equal to $1$ only when the base is $e$ is analogous to the fact that in the identity
$$
\frac d {dx} \sin x = (\text{constant}\cdot \cos x)
$$
the "constant" is $1$ only when radians are used rather than some other unit.
