A centered random variable $X$ with moment generating function (MGF) $M_X(t) := \text E[e^{tX}]$ is subexponential if $\log M_X(t) \leq ct^2$ for some $c > 0$ on some neighborhood $(-\delta, \delta)$ of zero.
Are there any examples of random variables with finite MGFs in some neighborhood of zero that are not subexponential? Relatedly, how quickly can the MGF grow near zero while still being finite, and, for a given growth rate, how to find a random variable with that MGF?