(I'm not a physicist.)

It seems from the little I've read that the "canonical" example (or maybe just easiest or first example everyone goes to) of a quantum system whose Hilbert space has 2 dimensions comes from thinking about the spin of a particle; the 2 dimensions comes from the spin-up and spin-down states being a basis.

My question is very simple: Is there a "canonical" (or easy or first) example of a quantum system whose complex Hilbert space is $n$ dimensional for arbitrary $n$?