This seems like a weird question, because the series has good convergence and there's no need to use other methods to estimate it for $N \to \infty$.
However, after seeing this question, I tried to come up with some approximation which could allow us to treat integrals containing this sum in the denominator.
Regardless of the particular application, I just wanted to ask if there's a way to approximate this sum using a finite combination of elementary or special functions?
Not counting the obvious $e^x-\sum_{k=0}^{N-1} \frac{x^k}{k!}$ of course.
Euler-Maclaurin formula doesn't seem very promising, because the resulting integral is even more complicated $$\int_N^\infty \frac{x^y}{\Gamma(y+1)}dy$$ And the derivatives containing various combinations of polygamma functions are also a pain to write.
If there's no better asymptotic expression than the sum itself, so be it.