Let $\sum u_n z^n$ denote the power series of $e^{1/(1-z)}$. As our radius of convergence is $1$, it follows that $u_n$ exhibits sub-exponential growth. On the other hand, $\{u_n\}$ must grow supra-polynomially, else transfer theorems like those found in Singularity Analysis of Generating Functions would then imply that the singularity at $z=1$ is regular. Heuristically, it would appear that $$u_n \sim \alpha n^{-3/4} e^{2\sqrt{n}},$$ for some $\alpha \approx .162982$. This opinion is echoed, without support, as a comment on the OEIS page for A000262. Note: The sequence considered therein is the generating function of $e^{z/(1-z)}$, which has $\mathbb{Z}$ coefficients after scaling $u_n$ by $n!$
How would one derive asymptotic results such as these?
(Edited for spelling.)
Note: I've posted my own solution in the answers below. In short, the saddle-point method applies.