Let $\operatorname{Li}_s(z)$ denote the polylogarithm function $$\operatorname{Li}_s(z) = \sum_{k=1}^\infty \frac{z^k}{k^s}.$$
Does there exists a closed form or a known function which generates the polylogarithm on powers of the order $s$? I mean How to formulate $G$ such that
$$G(x,z)=\sum_{n=0}^\infty \operatorname{Li}_n(z) x^n$$ or the exponential generating function for the polylogarithm $$G(x,z)=\sum_{n=0}^\infty \frac{\operatorname{Li}_n(z)}{n!}x^n$$ instead?
For example we know for $z=1$ and starting from a higher order, its generating function is Digamma, as shown in Generating functions and the Riemann Zeta Function
Supposing $|z|<1$ and that we have the right to reverse the order of the series we get : \begin{align} G(x,z)&:=\sum_{n=0}^\infty \sum_{k=1}^\infty \frac{z^k}{k^n} x^n\\ &= \sum_{k=1}^\infty z^k \sum_{n=0}^\infty \left(\frac xk\right)^n\\ &= \sum_{k=1}^\infty \frac{z^k}{1-\frac xk},\quad\text{for $x\not\in \mathbb{N}$}\\ &= \sum_{k=1}^\infty \frac{k\;z^k}{k-x},\quad\text{for $x\not\in \mathbb{N}$}\\ &= z\frac d{dz}\sum_{k=1}^\infty \frac{z^k}{k-x},\quad\text{for $x\not\in \mathbb{N}$}\\ &= z\frac d{dz}\left(z\,\Phi(z, 1, 1-x)\right),\quad\text{for $x\not\in \mathbb{N}$}\\ &= z\left(\frac1{1-z}+x\,\Phi(z, 1, 1-x)\right),\quad\text{for $x\not\in \mathbb{N}$}\\ \end{align} using the Lerch zeta function $\displaystyle\;\Phi(z, s, \alpha) := \sum_{n=0}^\infty \frac { z^n} {(n+\alpha)^s}$
Your second generating function appears not so easy : \begin{align} H(x,z)&:=\sum_{n=0}^\infty \frac{\operatorname{Li}_n(z)}{n!}x^n\\ &= \sum_{k=1}^\infty z^k \sum_{n=0}^\infty \frac 1{n!}\left(\frac xk\right)^n\\ &= \sum_{k=1}^\infty z^k \exp{\frac xk}\\ \end{align} but it seems that Gary already provided an answer!