I'm reading SSH right now, and they've come up with an example where you're behind n the turn, but all 44 remaining cards will give you an out to either a split pot or a win. So I guess 44 outs is the most mathematically possible. HEre's the example, lifted directly from a hand quiz on counting outs in SSH:

Your Hand: Ad 2d
Your Opponent: As 3s

Board:Ac 8d 8h 2h

A: You have two outs to win (2 deuces) and 42 outs to tie (every other card). The pair of eights counterfeits your bottom pair, so you have aces and eights with a deuce. Your opponent has aces and eights with a trey, so he is currently "ahead." A deuce o the river gives you deuces full of aces, but it does not improve your opponent. Any other card counterfeits your opponents kicker - he cannot win. Even a trey counterfiets his kicker, since it will leave you with aces and eights with a trey.