How many rounds fired though a new chrome moly barrel before it is considered broken in?

That is a loaded question for which there is no answer; it depends on which barrel, how smooth it is, how it was rifled, the bullets you are using. Ask the maker. Some say shoot, clean, shoot, for 5 rounds. Then do the same but shoot 5, clean. Then do the same for ten rounds. Every maker and shooter has his own method.

It depends on the quality

A lapped barrel will need little to no break in time

But then again a production factory hammer forged $40 Remugerchester might need 100/200 rounds and still not be smooth

That being said....smooth and or easy to clean doesn't necessarily mean accurate

"Remugerchester", good one

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As a rule, button rifled barrels seem to change very little, if at all. They just slowly wear out. Some times they will need to foul in, but that only takes three or four shots. There are of course always exceptions though.
Cut rifle barrels on the other hand which haven't been lapped can do some very funky stuff for the first 100 rounds or so. I have seen them start out as psychos and then over time settle in to become honest members of society. I have also seen them start out as one hole wonders and then for some reason they seem to lose interest and spend the rest of their lives shooting around the outside of snuff tins, sniffing coke and robbing likker stores.
After about 5 boxs of ammo most rifles generally get past the bed wetting and teething stage and they get more predictable.

What about hammer-forged barrels?

I have always said on a custom lapped barrel, the only thing it needs is getting rid of the fluff in the new chamber. It is in the throat leade area from chambering. I take a good quality bronze brush with a small amount of 0000 steel wool wrapped around it. Take your electric drill motor and massage the throat for about 10 seconds. Your barrel is ready.

Crap you mean for the last 45 years I needed to break in my barrels??????

Quality barrel I'm with Butch a little clean up and I go shooting. Factory hammer forged why bother.

To be honest I can't remember a cut barrel. I know I did the 5 clean 5 for my first barrel. Didn't the second saw no diff so stopped bothering.

I'm with Butch on this, clean up the throat and they seem to be ready to rock. Factory tubes get treated to Final Finish bullets.

Like butch said on good barrels you break in the throat and nothing more.

on anything else don't worry because it won't shoot any better then it is right now