Problem
Let $F_{n,p}(x)$ be a family of smooth, monotonically increasing functions $F_{n,p} : [0,\infty) \to [0,1]$ indexed by an integers $n,p$.
The $F_{n,p}$ satisfy the recursion relation $$ F_{n,p}(x) = \begin{cases} \displaystyle 1 & \quad (n = 0) \\ \displaystyle \left( \int_0^x \mathrm{d}y \, \mathrm{e}^{(y - x)} \,F_{n-1,p}(y)\right)^{p} & \quad (n > 0) . \end{cases} $$ For the cases $p=1$, calculating the $F_{n,p}$ is straightforward and we find $$ F_{n+1,1}(x) = \frac{1}{n!} \int_0^x \mathrm{d}y \, y^{n} \mathrm{e}^{-y} \quad (n\geq 0) $$ which in the limit of $n \gg 1$ tends to $$ F_{n,1}(x) \to \frac12 \left[ 1 + \mathrm{erf} \left( \frac{x - n}{\sqrt{2n}} \right)\right] $$ in the sense that $$ \lim_{n \to \infty} F_{n,1}\left(\sqrt{2n}z+n\right) = \frac12 \left[ 1 + \mathrm{erf} (z) \right] $$
I would like to find similar limiting forms for $F_{n,p}$ for the regimes $p \gg n \gg 1$ and $n \gg p \gg 1$. Any help is very much appreciated.
Context
As context I note that each of the $F_{n,p}$ is a valid CDF, and this problem has an interpretation as one regarding the composition of independent random variables.
Specifically, consider a Cayley tree with coordination number $p+1$, and $n$ shells of vertices (not including the root) so that there are $N_d = p^d$ vertices at a distance $d$ from the root (and a total number of vertices is $N= \sum_{d = 0}^n p^d $). Let us associate an iid exponentially distributed random variable $x_e$ with each edge $e$ of the tree. Then $F_{n,p}$ is the CDF of a sum of the $x_e$ along a path from the root vertex to the boundary, maximised over possible paths to the boundary.