From search result depths, I stab at thee!

When a proton or neutron (or any baryon) are collided with at energies high enough to 'smash to bits', we should start introducing quantum chromodynamics terms. Essentially, the moment you have enough energy to split a baryon, you have to overcome the colour confinement (strong force effect translating to energy binding quarks together). The energy required to accomplish that is enough to create antiquarks of complementary colours, so instead of loose quarks - aka free quarks, that can't exist below the wishy-washy string theory gluon soup energies - you get new particles. Those also carry their share of initial kinetic energy, which can produce further particles. The only 'limit' are total energy and conservation laws (energy, momentum, quantum numbers, the whole shabang). That's why you get those cascades from even a simple proton-proton collision.

As to what could happen when one hits you in the nose, Anatoli Bugorski could point you in a general direction. I'd back-of-the-envelope estimated the collision to be less likely to occur by about four orders of magnitude because it's a single particle through a nose, not a packed beam through skull, but that's a good starting point for any what-ifs.