One example in fact stares us in the face. One learns in advanced Quantum Mechanics that fermions have to be ``quantized'' by anti-commutators, i.e., with rules involving AB+BA rather than AB-BA. This has simple connection to the Exclusion Principle. Further the field operators one introduces to represent a collection of fermions have no classical limit. Only quadratic expressions in these operators have classical limits, for example electric current for electrons is a quadratic expression in the underlying field operators. In dealing with the mathematics of such operators one does introduce classical numbers, but then they are Grassmann or anti-commuting ``numbers'', not accessible on the number line or the complex plane.
There are other bizarre situations thrown up by Quantum Mechanics. The quarks that constitute the neutrons and protons cannot in principle be observed isolated. This is because their self-interaction is so strong that it modified the ground state of the system. Unlike a potential well which particles might roll into, here it is like a phase transition. The phase with a pair of free quarks is infinitely higher in energy than the one where the pair is bound up in a slew of gluons and quarks.
Perhaps we are yet to uncover a large number of situations in which the newly discovered principles, only a century old, can be manifested.