Here's a picture of my simple application:
As you can see, I'm making chinese checkers.
When the mouse is over the pieces, they turn white. If you click and drag the pieces, they'll move around the board. They have no texture but the default Object3D texture assigned, plus the addtional color.
Anyway, this seemed a good thing to start with because it has only a few rules, it has hex tiles which are good for war simulation games later, and I can do a bit of AI with it. The model for the board has very few triangles and I'm still learning how to map the texture correctly, but it's not so important to me right now. I'm more interested in the game play, what little there is. I had a lot of the game logic done a while ago, now I'm finding problems with my hex tile representation. I don't spend very much time on this, so it's going slowly.