Twenty-five. He could avoid pitching in the ninth inning if his team lost. But in order to lose he must have allowed at least one run, and that would require at least one pitch. Twenty-four additional pitches produced three outs in each of eight innings, for a total of 25 pitches.

02/06/2012 UPDATE: I found this problem in the Litton Problematical Recreations series, but as many of you have pointed out, the answer is wrong. Technically there are at least two ways in which a pitcher can complete a game without throwing a single pitch:

— Rule 8.04 says that if a pitcher fails to deliver the ball within 12 seconds of receiving it, the umpire can call a ball. A very dilatory pitcher could put 27 men on base in this way and then pick them all off.
— If the pitcher plays right field, he won’t pitch at all. :)

Thanks for everyone who wrote in about this.