What is this crud in my shower valve, and how can I prevent it?
These are just educated guesses since I cannot feel/smell these white deposits:


My best guess is that your dip tube in your water heater disintegrated.  This is a very common problem especially if your water heater is 10-25 years old.  You simply need to replace dip tube and then go to all valves and faucets and clean them.
Calcium deposits.  I doubt this is it because I have seen it so many times that I think would be able to tell.  The pieces would look more like rocks.  They would have a little bit of greenish white tint.  There is no fix for this beyond filtering all incoming water.
Chlorine tablets.  Your water source puts chlorine tablets in the water.  If you are near the source the tablets may get into your water without fully dissolving. Your issue does not look like this but throwing it out there.
Rubber hoses.  Any kind of rubber hose that you have in your plumbing system.  Water heater, under sinks, for toilets...  These can break down over time and actually the outcome is a lot like yours, except most hoses are usually black.