If you by rings mean world series rings, then it is Yankee great Yogi Berra, with 10 rings.
Yogi Berra has won 12 World Series rings as a Yankee. He won 10 as a player, which is the most in baseball history, and 2 more as a Yankees coach.
Derek Fisher as 5 rings
Tom Brady, with three.
Lawrence "Yogi" Berra won 10 World Series championships with the Yankee's as a player. He has a total of 13 rings, with 3 of them coming as a manager.
Jorge Posada is second on the Yankees active hit list behind Derek Jeter.
Manny Ramirez is the top active non-Yankees player with the most postseason appearances with 111 games played.
Derek Fisher & Kobe Bryant, both with 5