Assuming OBR, an UNLIMITED number!
Get 2 outs by K's or any other manner. Then for as many batters as you got, do an uncaught 3rd strike! Then, have F2 fail to make a play correctly.
In each case, F1 gets credit for a K, and F2 gets an error. But the game (and the inning) plays on!
From Wikipedia, four strikeouts recorded in one half-inning has happened 60 times in MLB play, the last being done by Yovani Gallardo of the Brewers against the Reds on September 17, 2011. Chuck Finley of the Angels and Indians actually did it three times in his career, twice in 1999 and then again in 2000.
Five strikeouts in one half-inning has never happened in a regular season MLB game. Joe Niekro did it in a spring training game while pitching for the Astros. There were at least two recorded instances of it happening in MiLB, by Mike Schultz of the Lancaster JetHawks in 2004, and by Garrett Bauer of the Rockford RiverHawks in 2008.
One interesting feat that occurred involving a four-strikeout inning: In a 2003 game between the Houston Astros and the New York Yankees, Octavio Dotel's four strikeouts in one inning was part of a string of eight consecutive Ks the Astros recorded. In the bottom of the 7th, after Hideki Matsui grounded out against Brad Lidge, Lidge got Todd Zeile and and Raul Mondesi to go down swinging. In the bottom of the 8th, Dotel recorded strike-outs of Juan Rivera, Alfonso Soriano, Derek Jeter, and Jason Giambi. Soriano reached on a wild pitch. Billy Wagner then closed out the game by striking out Jorge Posada and Bubba Trammell, and then getting Matsui to ground out. The Astros won the game, 8-0 on a no-hitter involving six Astros pitchers!
BTW, there has never been, as near as I can tell, an MLB game with a full inning where more than six strikeouts have been recorded.