Talking with a friend about them.
Does the chrome wear out of them? How many rounds does it take?

Yes. It takes 125,000 rounds to wear out.
I am familiar with chrome plated bores. The M256 tank cannon barrels are chrome plated, smooth bore, just like a big shotgun. They are plated in vertical vats in the floor, at Watervliet Arsenal. They have a life of 1200 shots, with Sabot ammunition; more with target practice rounds. But they operate at 90K psi, and use 26 pounds of propellant at a velocity of 4800 FPS. They cost $100K each, to make.

Lots of competitive shooters have 250,000+ rounds through their shotguns and I've never heard of anyone complaining about the chrome being an issue in the bore.

I have at least 150K through my K-80 and the bores look fine.

Getting the action to last that long on a shotgun is a bigger issue than the chrome be scrubbed away by the plastic wads. The higher end guns can have the actions rebuilt indefinitely, but I couldn't get 150K rounds out of the 680 series Beretta's because over-sized parts are only available to rebuild them four times.

I guess the question would best be asked of somebody like Briley or Cole who regularly see the most heavily shot competition guns.

I can't guess at how many rounds I have put through my chrome barreled Miroku O/U shotgun since purchasing new in 1984 and using for both clay target competition and gamebird with both lead and steel shot. Bores are still shiny smooth without blemish just like new.
If chrome bores are shiny then they will not be wearing. Unless abused I don't imagine you would wear quality shotgun chrome bores in a lifetime.