I think the elements of the base need to be enumerable (proof needed but it feels natural), and transcendental numbers are not enumerable (proof also needed).

Base pi: https://en.wikipedia.org/wiki/Non-integer_base_of_numeration...

Base e: https://en.wikipedia.org/wiki/Non-integer_base_of_numeration...

> I think the elements of the base need to be enumerable (proof needed but it feels natural)

Proof of what? Needed for what?

The elements of the number system are the base raised to non-negative integer powers, which of course is an enumerable set.

> transcendental numbers are not enumerable

Category mistake ... sets can be enumerable or not; numbers are not the sort of thing that can be enumerable or not. (The set of transcendental numbers is of course not enumerable [per Georg Cantor], but that doesn't seem to be what you're talking about.)

I think your parent comment was speaking of a "base-$\alpha$ representation", where $\alpha$ is a single transcendental number—no concerns about countability, though one must be quite careful about the "digits" in this base.

(I'm not sure what "the elements of the base need to be enumerable" means—usually, as above, one speaks of a single base; while mixed-radix systems exist, the usual definition still has only one base per position, and only countably many positions. But the proof of countability of transcendental numbers is easy, since each is a root of a polynomial over $\mathbb Q$, there are only countably many such polynomials, and every polynomial has only finitely many roots.)

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