There is no limitation to the number of innings a Baseball game can be played. In theory, the game could go on forever as long as the teams are tied at the end of an inning. The longest professional game I have heard of is a 33 inning game between Pawtucket and Rochester of the International League (minor leagues) in 1981. The longest MLB game was 26 innings between Brooklyn and Boston in 1920.
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That is a tough question to answer because of variables that would limit how many innings the pitcher could pitch, such as pitch counts and how good the pitcher was pitching and, in the National League, whether the pitcher is coming up to bat in an inning and the manager decides to pinch hit for him.
MLB starting pitchers pitch every 5 games which means they would pitch no more than two games in one week. The most innings pitched in MLB in 2010 was 250 2/3 by Roy Halladay in 33 starts. This averages out to 7 3/5 innings per gsme and 15 1/5 innings every two games.
So, the pitcher that pitched the most innings in MLB in 2010 averaged between 15 and 15 1/3 innings per week.
Most Major League Baseball teams use a five man pitching rotation. That means that a starting pitcher will usually pitch every five games.