The ongoing puzzle at @Resistance2018 (promoting the new Terminator movie) is probably the first -- and excellent -- attempt at advergaming on Twitter, but I'm anxious to see how people will explore the potential of this new medium further. While Resistance2018 is not really an ARG, it's easy to see how Twitter will become another tool for ARG gamemasters, like those behind the new rumored (but possibly unofficial) Lost game unfolding over at @Simeonhobbes.
@eamobile, a community manager for EA, experimented with a battleship game on his account.
I really like the idea of making Assassin-like games (@assn).
I found a couple of people who are trying to use Twitter as a Zork engine, for example: @zorker and @zorkgame. Once you get this technology going, you can use it for a bunch of other stuff, like, I don't know, new employee orientation (I always wanted to do an employee-based text adventure).
Some other people are trying to play a TwittRPG. And a @textadventure.
Also, there are a couple of related articles on Mashable and AdFreak's TweetFreak, too.