Function != Computable Function / general recursive function.
That's my point - computable functions are a [vanishingly] small subset of all functions.
For example (and close to our hearts!), the Halting Problem. There is a function from valid programs to halt/not-halt. This is clearly a function, as it has a well defined domain and co-domain, and produces the same output for the same input. However it is not computable!
For sure a finite alphabet can describe an infinity as you show - but not all infinity. For example almost all Real numbers cannot be defined/described with a finite string in a finite alphabet (they can of course be defined with countably infinite strings in a finite alphabet).
Non-computable functions are not relevant to this discussion, though, because humans can't compute them either, and so inherently an AGI need not be able to compute them.
The point remains that we know of no function that is computable to humans that is not in the Turing computable / general recursive function / lambda calculus set, and absent any indication that any such function is even possible, much less an example, it is no more reasonable to believe humans exceed the Turing computable than that we're surrounded by invisible pink unicorns, and the evidence would need to be equally extraordinary for there to be any reason to entertain the idea.
Humans do a lot of stuff that is hard to 'functionalise', computable or otherwise, so I'd say the burden of proof is on you. What's the function for creating a work of art? Or driving a car?