The complexity class $\newcommand{\sharpp}{\mathsf{\#P}}\sharpp$ is defined as

$\qquad \displaystyle \sharpp = \{f \mid \exists \text{ polynomial-time NTM } M\ \forall x.\, f(x) = \#\operatorname{accept}_{M}(x)\}$.

It is known that $\sharpp$ is closed under addition, multiplication and binomial coefficient. I was wondering if it is closed under power. For example, we are given a $\sharpp$ function $f$ and another $\sharpp$ function $g$. Is it true that $f^{g}$ or $g^{f}$ are $\sharpp$ functions as well?