By "natural", I mean a situation where $e^e$ appears non-trivially. For example, when analyzing a dynamical system or a model of some physical process.
Apparently it is still an open problem whether $e^e$ is even irrational (see Is $e^e$ irrational?).
In the study of random graphs, you get not just $e^e$ but the entire function $e^{e^x}$ naturally. In several places, but for instance:
Choose a random graph $G$ on $n$ vertices adding an edge between each pair of vertices independently with probability $p = \frac{\log n + c}{n}$. Then $$\lim_{n \to \infty} \Pr[G \text{ is connected}] = e^{-e^{-c}}.$$
(The reason we choose $p$ to have this particular form is that the limit will just be $0$ or $1$ if $p$ is much smaller or much larger, respectively.)
The rough idea is that lots of things in random graphs have a Poisson limiting distribution, and if $X$ is Poisson with mean $\lambda$, then $\Pr[X=0] = e^{-\lambda}$. In the case of connectivity, there are two separate Poisson distributions at work: the distribution of the degree of a given vertex, and the number of vertices with degree $0$. The $e^{-\lambda}$ from one of them feeds into the $e^{-\lambda}$ from the other.
