How high does a foul tip have to go for it to be an out when caught by the catcher?
There is NO height requirement that a batted fly ball must travel, anywhere on the playing field, before it can be caught for an out...just like any other fly ball.
It's not head height, or shoulder height, or six feet high, etc, etc, etc...
The key is this: What path did the ball take off the bat? "Sharp and direct", like in a straight line to the catcher, OR, with "perceptible arc" (ie: anything other than a straight line)?
The "sharp and direct" hit can NEVER be caught as a fly ball for an out. It might become a foul tip (a strike) or a foul ball, but it's not treated as a batted fly ball.
The ball that comes off the bat with "perceptible arc" is considered to be a batted fly ball. It can be caught for an out- just like any other batted fly ball.
That a batted ball must travel some given height before it can be caught for an out is a long held RULE MYTH. I've seen this same question on other discussion boards about a half dozen times already this season, plus had it come up a couple of times in my games.
The source of confusion seems to be that people read the rule (or, the old rule, for those associations that have changed their rule) and misinterpret the definition of a "foul tip". The rule says if the ball goes higher than the batter's head,
it cannot be a foul tip. It DOES NOT say "the ball can't be caught as a fly ball for an out". The "above the batter's head" part of the rule applies ONLY to foul tips- that is, batted balls that go in a straight line, sharply and directly to the catcher.
The misconception seems to be that people assume if a foul tip can't go higher than the batters head, then the opposite must be true for a batted fly ball- that it must go higher than the batter's head to be caught. That simply is not, and never has been the case...despite rule myths to the contrary.