It is relatively straightforward to show that when an analytic function has a simple pole, then the coefficients $a_n$ of the Taylor series asymptotically follow
$$ \left|a_n\right| \sim O\left(z_0^{-n}\right) $$
as $n \to \infty$, where $z_0$ is the location of the pole relative to the point of Taylor expansion. There are similar results (with binomial coefficients involving $n$) for higher-order poles.
I think that I have seen an argument that the converse of this is also true, i.e., if $\left|a_n\right| \sim c^{-n}$ for some $c$, then $c$ is a simple pole. It is intuitively clear that near $z=c$, the series becomes very close to the series for $\frac{\text{1}}{1-\frac{z}{c}}$ (ignoring a constant factor), so I suspect that this is correct.
Assuming that the converse is true, then all of this suggests that we should be able to derive rigorous bounds on the asymptotic behavior of $a_n$ for entire functions. At the least, $a_n \sim o\left(c^n\right)$ for any $c$. On the other hand, we can look at standard entire functions such as $\exp(z)$ and conclude that it's possible for an entire function to have $a_n \sim \frac{1}{n!}$. It's not immediately obvious what is possible in the region between these two behaviors (exponential decay and inverse factorial decay).
What actual bounds are known here? Does anyone have a reference that discusses these issues?