A "non axiomatic" approach would not be mathematical, unless I misunderstand what you're asking. Ultimately everything rests on the axioms. However, there are various ways to characterize the reals, such as
- the unique complete ordered field,
- the largest Archimedean field,
- the equivalence classes of Cauchy sequences in Q,
- the Dedekind cuts on Q,
- and weirder ones.
In classes I've taken, we've done the axiomatic approach (here are the field axioms, order axioms, and the least-upper-bound property, and take my word for it that only the reals satisfy this) and the method of Dedekind cuts. But I'm sure that for every valid, reasonable method, there's some class somewhere that teaches it.