Every time I teach NP-Completeness, students ask "are there any problems that are known to not belong to NP?"
How would you answer? I usually give them an undecidable problem as an example, but this often doesn't turn out well: (a) if I give them the Halting Problem they think it's some dumb corner case, and (b) if I give them Diophantine Equations they don't see why it's not in NP (you can check solutions in poly-time... just plug them in! I have a hard time disabusing them of this approach.)
I'd like to give them something like QBF as an example, but there is no proven separation.