how many rounds do you need to run down a new barrel before it achieves it's best grouping?

depends on the barrel, here are shots 4,5, and 6 from a lilja 9 twist .284 barrel

RR

now you are bragging. great group. in reading some other post on this board i got the impression that you shouldn't expect your best groups for a number of rounds but no one ever said how many. in my case i am starting out with a new savage long range hunter in 300 mag.

it depends on a lot of things, how smooth was the barrel when you started, was it lapped, what other work is done to the action and stock, its not something you can put a number on.
RR

What RR said. Some barrels respond right away. Others have taken up to 200 rounds with a good elbow workout with J&B compound between sessions. Other factory barrels have been a waste of powder and lead.