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Example of a project: well the TODO list suggested in the other comment makes for a great example: it can be very small. But there could also be more original stuff, things which would not repeat anything out there. Another example would be a small iPhone arcade-like game. There are plenty such ideas, and that's the point :P Project Euler level of questions is precisely what I was thinking of (in terms of length), in fact, since working on some of its questions with a friend is a part of what made me think of the idea.

I really like your idea of Yahoo games style "programming room", but I think it'd be overkill in the beginning. I'm thinking of something more flexible, based on different (existing) technologies to coordinate the work, say simply IRC, SVN, some Wiki page, etc. It doesn't have to be a complex website: the functionality would be more at the level of getting people together and allowing them to comment on each other's work (ratings etc.). Where they go from there is suggested, but up to them.