If you can do so, you'd have proven Turing, Kleen, Church, Goedel wrong, and disproven the Church-Turing thesis.
No such example is known to exist, and no such function is thought to be possible.
> Turing machines (and equivalents) are predicated on a finite alphabet / state space, which seems woefully inadequate to fully describe our clearly infinitary reality.
1/3 symbolically represents an infinite process. The notion that a finite alphabet can't describe inifity is trivially flawed.
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).
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.
For starters, to have any hope of having a productive discussion on this subject, you need to understand what "function" mean in the context of the Church-Turing thesis (a function on the natural numbers can be calculated by an effective method if and only if it is computable by a Turing machine -- note that not just "function" has a very specific meaning there, but also "effective method" does not mean what you're likely to read into it).
I would reframe: the only way of showing that artificial intelligence can be constructed is by showing that humans cannot compute more than the Turing computable.
Given that Turing computable functions are a vanishingly small subset of all functions, I would posit that that is a rather large hurdle to meet. Turing machines (and equivalents) are predicated on a finite alphabet / state space, which seems woefully inadequate to fully describe our clearly infinitary reality.