There have been three pitchers to strikeout 20 batters in one game and one pitcher to strikeout 21 batters in a game. The three to strikeout 20 were Kerry Wood, Roger Clemens, and Randy Johnson. All three struckout all those batters in a nine inning game. The pitcher to strikeout 21 batters, Tom Cheney, needed 16 innings. Cheney's 21 strikeouts is the MLB record for strikeouts by a pitcher in a single game.
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Nolan Ryan struck out the most batters, but Reggie Jackson struck out the most at the plate.
In MLB, that was Tom Cheney of the 1962 Washington Senators.
Cheney struckout 21 Baltimore Orioles in a 16 inning, complete game performance on September 12, 1962 winning 2-1.