Having just returned from my trip to Paris (the 5th in the past decade), I truly think there are many less touristy points of interest that also offer fantastic travel experiences. After all, I only was able to visit more of those recently (having fully enjoyed the more "touristy" spots). It's not to undermine the top attractions, but also to offer new ideas to enjoy the city.
My choices are:
Pere Lachaise cemetery: I was fortunate enough to already include this in my itineary during my earlier trips. Couldn't rave more about this place... always had a great time... tip: go on a sunny day!
Saint Martin Canal: finally made it there this time and enjoyed a relaxing afternoon watching the boat passed through the canal.. quite interesting to watch. The area seems to be one of the up and coming neighborhoods that have the potential of being a cool place to stay/be in the near future.
Mouffetard Market: more of a local market experience v. the Rick Steve made famous Rue Cler. There are a lot more selections at better prices. I enjoyed it a lot.
Le Marais: probably still considered "touristy" but I highly recommend experiencing the neighborhood by foot and spend a morning or afternoon with no maps/tour books, etc. I wouldn't mind getting lost a bit in the Marais. Chances are, I'd come across interesting stores and cafes not mentioned anywhere in the tour books.