  A function $\mathbb{N} \to \mathbb{N}$ is near exponential if it is bounded
above and below by functions of the form $2^{n^c}$ for some $c > 0$. In this
article we develop tools to recognize the near exponential residual finiteness
growth in groups acting on rooted trees. In particular, we show the near
exponential residual finiteness growth for certain branch groups, including the
first Grigorchuk group, the family of Gupta-Sidki groups and their variations,
and Fabrykowski-Gupta groups. We also show that the family of Gupta-Sidki
p-groups, for $p\geq 5$, have super-exponential residual finiteness growths.
