Retrosheet announced this month that it has uploaded complete box scores back to 1901. Those additions have been incorporated into Baseball Reference's database, as well.
Both sites now have box scores for every game in American League history.
Retrosheet has also added partial play-by-play accounts for two additional years (1916 and 1917).
Baseball Reference's Game Finder tools can now search back to 1901. Batting and Pitching Event Finder searches now go back to and including 1916 (though not complete).