"He told me that in 1886 he had invented an original system of numbering and that in a very few days he had gone beyond the twenty-four-thousand mark. He had not written it down, since anything he thought of once would never be lost to him. His first stimulus was, I think, his discomfort at the fact that the famous thirty-three gauchos of Uruguayan history should require two signs and two words, in place of a single word and a single sign. He then applied this absurd principle to the other numbers. In place of seven thousand thirteen, he would say (for example) Maximo Perez; in place of seven thousand fourteen, The Railroad; other numbers were Luis Melian Lafinur, Olimar, sulphur, the reins, the whale, the gas, the caldron, Napoleon, Agustin de Vedia. In place of five hundred, he would say nine. Each word had a particular sign, a kind of mark; the last in the series were very complicated ... I tried to explain to him that this rhapsody of incoherent terms was precisely the opposite of a system of numbers."

(Borges, Funes the Memorious: https://www.sas.upenn.edu/~cavitch/pdf-library/Borges_FunesT... )

That's a tricky one. Should we be able to actually write down each and every digit? If so then you'll need to invoke some sort of infinite related concept to allow it to happen. I'm not a mathematician but:

Let the digit for one be a single pixel (the simplest mark I can make), two is two lots of one pixels etc. Now we need an infinite number of these.

I don't think we can change over to say symbols made up from pixels either, nor mess with multi dimensions to add extra "depth". In the end we still need an infinite number of pixels or pixel properties - we could mess with colours but that is simply adding dimensions.

We can conceive of base infinity but I don't think we can actually use it as such except symbolically. We can decree that a particular symbol or construction for a symbol represents a particular digit within base infinity that we can define by other means, and we can do that a lot but can we do it infinitely often? If we relax the physical representation requirement, then I'd say yes, otherwise no.

I've no doubt that this concept is well understood and dealt with already by the pros.

Is base 1 a valid implementation of base infinity?

Each unique digit pattern is a vertical line attached to the previous pattern.

Church numerals are kind of like this

I don't think it's possible to represent decimals this way though - 8.1 = "8 + 1/infinity" = 8.0 and so is any other number for the decimal

But does it even have a 'ones' place? What's ∞^0?