There is no limit to the number of fouls in the MLB.
The only rules about foul balls is if a batter attempts to bunt with two strikes and the ball is bunted foul, the batter is called out. But there is no limit to the number of foul balls a batter can hit when swinging at the pitch.
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If it's on a bunt attempt and you fouled on the third attempt you would be out. So if the batter attempted to bunt on all three strikes, and fouled all of them, (3) would be the limit.
You can technically have infinite foul balls (assuming they go out of the playing field and are not caught). The exception is if there are already 2 strikes and you foul the ball off but it goes right into the catchers mitt. Then you're out. Also, if they're are 2 strikes and you attempt a bunt and hit it foul then you are out.
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Above answer is almost correct...
A batter can hit an unlimited number of fouls, but a foul that is caught in the air is not counted as a foul. It's simply an out. Also, a ball that is hit straight back to the catcher who then catches it is not a foul...it is a "foul tip," and the ball is still alive, so that's not a foul either. And it doesn't matter if there are 2 strikes, or 1 strike or 0 strikes, or what the count is at all.
The first 2 foul balls that are hit and not caught are considered strikes. After that, there is no limit to the amount of foul balls that can be hit, as long as they're not caught.