Retrosheet has added much of the 1922 and 1923 NL seasons to their website. And, of course, 2008 is now available for both leagues.
Not every game has a play-by-play, but there are boxscores. You can check out the carnage in this Phillies-Cubs game back on August 25, 1922. (There is a formatting error though and the Batters Facing Pitcher column is unknown and the HRs allowed got shifted into it.)
Also, if you are so inclined you can download event files for the All-Star Game and the postseason.