On SABR-L, the listserv of the Society for American Baseball Research, Phil Lowry writes:
Thanks to Dave Black, we have now identified the second exhibition game to ever last 20 or more innings. It was a game April 10, 1898 in St. Louis, as the St. Louis Browns defeated the Milwaukee Brewers 12-11 in 20 innings and 3 hours 20 minutes. This is the 368th game to be found which has gone 20 or more innings.
Lowry is the author of Green Cathedrals, probably the definitive book on baseball parks. Nowadays he's focused on marathon baseball games such as these, and as is apparent from his identification of this one as #368, he's found and catalogued a whole lot of them!
Speaking of marathon games: what happens in the 20th inning of a 2006 Brewers game? Gabe Gross comes in to get a tough lefty? J.J. Hardy shows off that arm of his?