Quantum physics inherited its physical quantities from classical mechanics (energy, momentum, etc). Each quantity is associated with Hermitian operator, but there seems to be a lot more Hermitian operators than classic physical quantities! Granted, density operator can be viewed as an observable, however, it obeys restriction on its spectrum bounded between 0 and 1. Are there quantities besides classic ones?
Let us first note that, although all observables are self-adjoint, not all self-adjoint operators are observables. The obvious counterexample is the identity - something that "observes" the identity is just a silly thing like an instrument whose dial has been fixed to point at a "1", it has no physical meaning.
Another instance where non-observable self-adjoint operators appear is whenever we have superselection sectors1, which, by assumption, are invariant under observables. Any self-adjoint operator which does not have the sectors as invariant subspaces is not an observable, yet, such operators certainly exist.
Also, the usual quantization schemes, regardless of their rigor, do not assign to any self-adjoint operator a physical meaning. Rather, they take the classical phase space observables and "embed" them into the self-adjoint operators of the quantum theory.
Nevertheless, there are non-classical observables, and there is one that everyone knows: Spin. Spin does not exist in a classical theory, but emerges more or less naturally in the course of quantization when we are faced with constructing representations of the Lorentz group (or, non-relativistically, the rotation group).
1Superselection sectors occur when the representation of the algebra of observables is not irreducible, and are exactly the irreducible summands.
Every hermitian operator that appears in quantum mechanics (in some dynamics or symmetry of the system) is an observable in principle. Some of this operator is more easy to measure, and anothers is more (very more) difficult. When I say measure, I think in some mechanism that can correlate the hermitian operator of the system with the measure apparatus.
Operators that do not appears in dynamics and symmetry, are illness. How we can talk about things that don't have any connection to any thing.
Think about measure of the spin (electron). We can proceed with this because exist some sort of interaction that connect spins and magnetic field. Actually, for any generator of symmetry is expected that some interaction access this operator. Example, the momentum of a particle is a generator of symmetry, and we expect that this operator do something within, at least, a momentum of other particle. If our expectation failure we may have a very boring theory that do anything.