Currently the rules provide no provision for a character to progress beyond level 20.
With that said, they don't explicitly ban it, either, and as you've noticed, it would be simple to continue using a multiclass. The only things you would need to come up with houserules for would be proficiency, experience requirements, and multiclass spell slots.
Personally, I would rule that proficiency progresses according to the proficiency by CR table on page 8 of the Monster Manual. Multiclass spell slots could continue to gain 1 per level in the existing pattern (I can see it but I can't explain it) and experience requirements could continue to rise by 10000 every 2 levels.
Updated with the release of the DMG: The DMG includes suggestions for character advancement past the level 20 mark. This is the Epic Boons section on pages 231. The gist of it is that for every 30000 XP a character earns past level 20, they receive an epic boon. The list of Epic Boons is on page 232; they are essentially like feats, but considerably more powerful.