Ignoring the obvious safety issue of every student having an effervescent, volatile strong acid solution (which is tricky to handle safely even as an experienced chemist), there are a handful of incompatible chemical combinations with aqua regia that would require an immediate lab evacuation and call to the fire department. You'd need constraints and some serious vetting of the students' plans before you could let them touch the stuff.

I like the idea of open-ended experiments, but technique is so crucial to these things that even if they're right in theory, in practice they'd often get a negative result because of technical shortcomings of their technique. This is why you end up with cookbook experiments