I love the Musee des Arts Decoratif. Two years ago I had purchased a ticket in advance of my trip through FNAC only to arrive and discover that admission was free that day. (I had to return to FNAC and get a fluent French speaker to assist me in getting my money back.) I highly recommend this museum and their amazing temporary fashion exhibits. Find out when they are free and go then.
Paris parcs are free and amazing--Le Jardin du Luxemboug (watch the chess players playing); parc Monceau; parc Montsouris is just beautiful and has "talking benches"--it is a special project and fascinating; the Tuilleries; Palais-Royal garden is beautiful and great for people watching; Parc des Buttes-Chaumont, Parc de la Villette (home to Music Museum and Science Museum); La Promenade Plantee (the only above-ground promenade garden in the world--others are copying it; Parc Bois de Vincennes; Bois de Boulogne (especially Jardin d'Acclimation which is an amazing parc for kids).
Walk around and look for wall plaques of fascinating pieces of information about where famous people lived, etc.--Tolstoy residence (206 Rue de Rivoli) located by Angelina's. You can "bump" into these plaques all over the city.
Kiosque des Noctambules--ground level of Metro Palais Royal-Musee du Louvre. Decorative Metro entrance with glass colored balls.
Stravinsky fountains near the Pompidou Centre. Whimsical colorful fountains with many moving parts. It is the best fountain I have ever seen.
Le Viaduc des Arts--peruse the unusual artisan shops underneat La Promenade Plantee including the Musee de l'Argentiere.
Peek your head into Le Train Bleu (located inside Le Gare de Lyon train station) to see the amazing art nouveau decor.
Walk through the beautiful passageways including Galerie Vivienne.
Do the free 3-hour paris walk(s).
Walk through the Louvre de Antiquities across ffrom the Louvre. It is beautiful in there. While I was there last Sept '07 I saw a temporary exhibit on phonographs located inside and upstairs.
Musee de la Sculpture en Plein Air a series of sculptures at La Jardins...Tino Rossi near Ile de St. Louis.
During the summer there are lots of festivals in Paris
Visit Notre Dame--it is free (unless you want to walk upstairs and see the gargoyles and view--that costs money)
Visit churches including Sacre Coeur and many, many other amazing and beautiful churches in Paris
Walk in boulangeries and patisseries and look at the food art. By the way, the food art in these places tends to be very inexpensive for how beautiful it is.
Visit the tomb of the unknown soldier at L'Arc de Triomphe
Check out the La Defense area of Paris (west side of Paris) very different from the rest of Paris with interesting statues, etc.
Check out Bercy Village.
paris-walks.com tours are mostly 10E and fabulous for 2 hours of walking and learning
visit the cemetaries and explore. Cimetiere du Montparnasse is filled with literary greats, etc. I found a world champion chess player buried there too. Also Pere La Chaise is large and full of wonderfully famous people and stories (enter at the Metro Gambetta side so you are walking downhill in the cemetery). There are other great cemeteries too.
Visit the many outdoor street markets. Try free samples of goodies.
Check out the beautiful ceiling in Galeries Lafayette.
30 minute fashion show at Galeries Lafayette on Fridays at 3pm (need advance reservations acquired on line)
Louis Lepine Place near Notre Dame to see the beautiful flowers and on Sundays they add live birds and rabbits.
Walk around the various arrondissements as well as L'Ile St. Louis and Ile de la Cite.
Peek into Cafe des Deux Mouline, 15 rue Lepic near La Moulin Rouge to see where Amelie waitressed in her movie. (Don't eat here--it is suposed to be old and dirty)
Walk through some of the designer clothing stores as though you might actually buy something--eg Chanel, Dior, etc. They are amazing. So are the prices! I have bought makeup at Chanel though and it is fairly reasonably priced and really good quality.
Walk the Canal St.-Martin and watch boats go in and out of the various different-designed locks.
Check out the Artisan Well at Passy Square Lamartine. This is a chic area of Paris between 189 Av. Victor Hugo and 70 Ave. Henri Martin. Many people believe this water has healing powers. (Metro Rue de la Pompe)
Musee Edith Piaf (need appointment. call 01-43-55-52-72) donation only.
Drouot Auction House (world's oldest public auction house) to see real auctions going on. No charge.
There's a start.