This isn't something I've done; I haven't been able to try it. I don't know if anyone is willing to go through the trouble of playing games in such a slow, involved way when normal multiplayer games are available. But every so often, I get an idea in my head that I at least need to write out, so here it is.
Asynchronous gaming, as far as my knowledge goes, goes back to correspondence chess nearly a millennium ago. Special postcards have been made just for the purpose of playing chess over mail, with games lasting years at a time. I don't think I would want to go that far outside of some very special circumstance. I can't see myself playing the same game for years, taking months to come up with a countermove.
That said, the same idea can be applied to email. You can play games by sending moves in the form of files, which opens up certain possibilities that I found somewhat compelling. Board games, for example, can be played by recording moves in VASSAL and sending the resulting file to the next player. There are certain turn-based computer games that could probably be played like this too, like Civilization. I think there's actually a play-by-email option in the newer Civ games.
I also think it might be fun to play games simultaneously rather than in a multiplayer fashion. Single player games can be played in a sort of book club style, where everyone shares scores, secrets, strategies, and discusses the game for a week or two, moving on thereafter to a new one. I guess this isn't really a new concept. When any new game comes out, people naturally flock to forums and discuss it as they play. But maybe doing it more deliberately with older games would be cool.
But there's one idea that that I think is worth trying out. Years ago, I remember seeing options in emulators to record movies not to video files, but as series of inputs to be played back by the emulator itself. Retroarch still has this option, which you can access by pressing 'o' while playing a game. You can only record from within the program, however; you need use the command-line the to play the file back. The recordings are known as input movies, and they're saved as .srm.bsv in the Saves directory.
The command is
retroarch --bsvplay [path/to/bsvfile] --libretro [path/to/core] [romfile]. Once you start playing the file back, the game will play itself in just the way it was recorded. I think you can even save and load savestates while recording, and it plays back just fine. Take the recording and send it to the next player. When the playback is finished, the game simply continues to run, allowing the viewer to keep playing, recording gameplay and sending the recording back again.
Why would you want to do this? I'm not too sure, honestly. Maybe you have a pen pal who isn't awake at the same time as you are, but you want to play games one way or another. Maybe you want a little something to do during a break on busy days. Maybe you can have multiple games going with other people in a turn-based game club or something.
Anyway, this is something I've had in the back of my mind for quite a while now. If anyone else wants to try it, it would be cool to hear about the results.