I tried looking for something about this, but I couldn't seem to find anything so maybe I'm using the wrong terminology or maybe the idea is fundamentally flawed in a way I've not considered yet.
So just to give a concrete example to help clarify, turn the factorial number system into "n!-adic integers", then for instance I would expect something like this to converge for appropriate values of $x \in$ "$\mathbb{Z}_{n!}$",
$$\lim_{n \to \infty} \frac{x^{\varphi(n!)}-1}{n!}$$
I can clarify further, I realize I'm being quite terse and presumptuous but I'm more focused on getting on the right track for finding a mixed radix version of p-adics, the example is just a vague idea.