Keeping the constraint that the song has to have at most two chords, I think one shouldn't necessarily reject a tune just because it has more.

Theoretically, any diatonic melody, i.e., one that stays within a single scale or mode, can be harmonized with a single chord. The ultimate test is one's ears: if it sounds okay, then it is okay.

In nearly every case (in Western music), the two chords in a two-chord song will be the tonic and the dominant, e.g., C (tonic) and G7 (dominant). If it's the tonic and the subdominant, e.g., C and F, I would suspect that it might actually be the other way around with F as the tonic and C (or alternatively C7) as the dominant. One can always add the dom. 7th to the dominant, or leave it off. Again, for both points, one's ears are the ultimate test.

One might have something like Am G7, which would be the relative minor of C and the dominant of C. More common would be Am and E7 (i and V7 of A minor).

In short, one should feel free to experiment. You can't break anything and the worst that can happen is that it will sound awful.