Here is a nice challenge: Just like we have arbitrary precision integers (allocating more memory as needed) and arbitrary precision "fractional numbers", design a library providing arbitrary precision "computable numbers".
The other obvious representation is that of first order formulae over the theory of real fields, but then you need to check equivalence of arbitrary formulae. You ultimately need a limit operation to be included and now you're sunk. You'll also probably wish you were implementing the complex numbers instead pretty quickly.
I believe they call those "programming languages"
Programs written in programming languages aren't guaranteed to terminate. Computable numbers, however, are all real numbers that can be represented to arbitrary precision by a computation that terminates. Wouldn't that mean we don't need a fully fledged programming language to have the capability to represent all computable numbers?
Somewhere in there is the idea we just need a program that determines whether an arbitrary program terminates or not...