Montmartre, the Pere Lachaise Cemetary, Luxembourg Gardens, the Quartier Latin, Musee d'Orsay, the Catacombs, Cruise on the Seine river, Paris Plage (beach in the middle of Paris), Centre Pompidou...
Let me add the catacombs are simply a must see, they're unbelievable. Lots of stairs, though. Pere LeChaise is a good recommendation as well. See the graves of Chopin, Jim Morrison, Oscar Wilde, Edith Pilaf, many others.
- pop into St. Eusach church on any sunday at 5:30 and hear a free organ concert.
- go to the 19th/20th, walk through Parc de Butte Chaumont. Very few tourists there. The landscape and views are incroyable.
- go to the Pantheon, and see the tombs of many of France's great men. I usually pay homage to Jean Moulin and Volataire when I go.
- Speaking of tombs, go to Les Invalid and see Napoleon, Foch, and others
- Do you like modern art? Check out Palais de Tokyo to avoid the crowds at the Pompadou.
- A fabulous 19th century collecter left his home and its contents to the State. Its now the Musee de Camando, in the 8th (i believe). The family had a tragic ending in WW-II, but they left behind some great stuff.
- The Musee des Arts e Metiers in the 3rd is a great place to spend an afternoon, especially if you like technology and mechanical things. Its a shortish walk from the Pompadou, straight up Rue St. Martin.
- Take a boat ride down the Canal St. Martin. Its fascinating.
- Speaking of, one of my favorite little restaurants is right by the canal, its called Le Sporting. If you go you'll be the only tourist there. Its a great little neighborhood, tucked behind the Place de la Revolution.
- If you're at all into music, go check out the museum at the Cite de la Musique. Fantastic old instruments, and lots of orignal scores (its humbling to see something written in Bach's own hand)
Well thats a quick brain dump. If I think of other things I'll post them.