You can have 3 sacrifice flies in an inning if one of the sacrifice flies included an error that prevented the team from recording an out. This is similar to getting 4 strikeouts in an inning, where a missed third strike allows one of the batters to reach base on the strikeout. There have been 4 times where a team had 3 sacrifice flies in one inning. Strangely two of those were by the Yankees in 2000, both with Andy Pettite pitching for the Yanks, and one by the Mets against the Yankees in 2004. The other time was by the White Sox in 1962. In that August 19 game Angel left fielder Ron Gant dropped Jorge Posada's fly ball with a runner on third. Posada was credited with a sacrifice fly and an RBI, and he reached base on Gant's error.
In 2011, Russ Canzler played in 3 games, batting in all of them. He had 3 at bats, getting 1 hits, for a .333 batting average, with 0 sacrifice hits, 1 sacrifice flies, and 1 runs batted in. He was walked 1 times. He struck out 1 times. He hit only singles.
In 1956, Howie Pollet played for multiple teams. In total he played in 31 games, batting in all of them. He had 9 at bats, getting 3 hits, for a .333 batting average, with 2 sacrifice hits, 0 sacrifice flies, and 1 runs batted in. He was walked 1 times. He struck out 3 times. He hit only singles. In 1956, Howie Pollet played in 12 games for the Chicago White Sox, batting in all of them. He had 8 at bats, getting 3 hits, for a .375 batting average, with 1 sacrifice hits, 0 sacrifice flies, and 1 runs batted in. He was walked 0 times. He struck out 3 times. He hit only singles. In 1956, Howie Pollet played in 19 games for the Pittsburg Alleghenys, batting in all of them. He had 1 at bats, getting 0 hits, for a .000 batting average, with 1 sacrifice hits, 0 sacrifice flies, and 0 runs batted in. He was walked 1 times. He struck out 0 times.
In 1969, Danny Breeden played in 3 games, batting in all of them. He had 8 at bats, getting 1 hits, for a .125 batting average, with 3 sacrifice hits, 0 sacrifice flies, and 1 runs batted in. He was walked 0 times. He struck out 3 times. He hit only singles.
In 1942, Charlie Fox played in 3 games, batting in all of them. He had 7 at bats, getting 3 hits, for a .429 batting average, with 1 sacrifice hits, sacrifice flies, and 1 runs batted in. He was walked 1 times. He struck out 2 times. He hit only singles.
In 2000, Nelson Figueroa played in 3 games, batting in 2 of them. He had 3 at bats, getting 1 hits, for a .333 batting average, with 1 sacrifice hits, 0 sacrifice flies, and 0 runs batted in. He was walked 0 times. He struck out 1 times. He hit only singles.
In 1954, Bob Chakales played in 3 games, batting in all of them. He had 3 at bats, getting 1 hits, for a .333 batting average, with 1 sacrifice hits, 0 sacrifice flies, and 0 runs batted in. He was walked 1 times. He struck out 0 times. He hit only singles.
In 1981, Larry Christenson played in 20 games, batting in all of them. He had 30 at bats, getting 3 hits, for a .100 batting average, with 3 sacrifice hits, 1 sacrifice flies, and 1 runs batted in. He was walked 1 times. He struck out 16 times. He hit only singles.
In 1923, Sarge Connally played in 3 games, batting in all of them. He had 3 at bats, getting 1 hits, for a .333 batting average, with 1 sacrifice hits, sacrifice flies, and 0 runs batted in. He was walked 0 times. He struck out 1 times. He hit only singles.
In 1915, Dick Crutcher played in 14 games, batting in all of them. He had 13 at bats, getting 3 hits, for a .231 batting average, with 1 sacrifice hits, sacrifice flies, and 1 runs batted in. He was walked 1 times. He struck out 3 times. He hit only singles.
In 1936, Pat Ankenman played in 1 games, batting in all of them. He had 3 at bats, getting 0 hits, for a .000 batting average, with 1 sacrifice hits, sacrifice flies, and 0 runs batted in. He was walked 0 times. He struck out 3 times.
In 1976, Mike Marshall played for multiple teams. In total he played in 54 games, batting in all of them. He had 11 at bats, getting 1 hits, for a .091 batting average, with 1 sacrifice hits, 1 sacrifice flies, and 1 runs batted in. He was walked 4 times. He struck out 3 times. He hit only singles. In 1976, Mike Marshall played in 30 games for the Los Angeles Dodgers, batting in all of them. He had 5 at bats, getting 0 hits, for a .000 batting average, with 0 sacrifice hits, 1 sacrifice flies, and 1 runs batted in. He was walked 3 times. He struck out 2 times. In 1976, Mike Marshall played in 24 games for the Atlanta Braves, batting in all of them. He had 6 at bats, getting 1 hits, for a .167 batting average, with 1 sacrifice hits, 0 sacrifice flies, and 0 runs batted in. He was walked 1 times. He struck out 1 times. He hit only singles.