My favorite thing to do in Paris is my favorite thing to do most anywhere; ie. just hang-out, eat, meet people and hang-out with them, take day trips, etc., etc...
My brother's neighborhood in Paris is one of the best and you can walk to cool restaurants, great video rental places, markets, car dealers and every great store on earth. To say that it's a world-class city is a major understatement. I might be going in a couple weeks if the planets line-up properly.