Today, Chris brings up Runescape as an example of a Java app people actually use. I have to say, RS is pretty cool, and I do know some people that play it fairly regularly — it is great for the non-hardcore-EQ-geek crowd to get their MUD on.
Another one I would like to point out, though is Puzzle Pirates. It is the MMO expression of the casual games phenomena and really quite clever. Not only is is written in Java, but Three Rings has open-sourced their game framework code and even have a pretty clever offering called Game Gardens which is rather like the Second Life of casual games — create your networkable puzzle/board game and they will host it on their servers.