Comment by tromp

Turing Machines and Lambda Calculus can only output insanely large numbers by building those numbers from scratch using their Turing completeness. So while lambda calculus can output something exceeding Loader's Number, it needs well over a thousand bits to do so. What I mean by "geared toward outputting insanely large numbers" is saying: I define a language in which the 1-bit program "0" outputs Loader's Number. That is obviously cheating.

There is unfortunately no mathematically rigorous way to define what is cheating, so it seems unreasonable to ask me for that.