Somehow I have missed this thread until today. Glancing through it I don't think there has been any mention of the many free museums (other than the Carnavalet) owned by the city of Paris. Many of them are excellent. Here's a selection:
Musée des Beaux Arts de la Ville de Paris, housed in the Petit Palais, av. Winston Chuchill, 8ème. The collection is splendid, the building has just been completely renovated and is worth seeing, and the interior garden overlooked by a café, is lovely.
Musée d'Art Moderne de la VdeP, housed in the Palais de Tokyo built for the 1937 Exposition Universelle, av. du Président Wilson, 16ème. The collection is excellent and the space well thought out.
Musée de la Vie Romantique, housed in the Hôtel Scheffer-Renan, rue Chaptal, 9ème. THe "New Athens" neighbourhood was very fashionable in the early 19th century and was home to many artists, writers and musicians. The museum's collection includes works of art and furniture from the period. Much of the collection is associated with George Sand who lived nearby.
Musée Cognacq-Jay, housed in a Marais mansion, the Hôtel Donon, rue Elzévir, 3ème. The splendid collection of 18th century painting and sculpture, furniture, ceramics ad objets d'art, was amassed by the husnband and wife who founded the Samaritaine department store.
Musée Cernuschi, housed in a 19th century mansion close to the lovely Parc Monceau, av. Velasquez, 8ème. The collection of Asiatic, especially Chinese, art is among the finest in Europe.
Le Mémorial du Maréchal Leclerc de Hauteclocque et de la Libération de Paris - Musée Jean Moulin (Don't blame me. I didn't name the place!). can be found in the Jardin Atlantique above Gare Montparnasse, 14ème. It is not easy to find. Through two emblematic figures the museum traces the history of French resistance in WW2 leading up to the liberation of Paris in August, 1944.
These are just a few. There are also, among others, the Maison de Victor Hugo in the Place des Vosges, the Maison de Balzac in the 16ème, the Musée Bourdelle in the 15th, and the Musée Zadkine in the 6th, all worth a visit. Google them if you are interested but be aware that most of these museums have a very limited web presence.
Note that while admission to the permanent collections at these museums is free there is usually a charge for special exhibitions.
PS: Before anybody gets all over my case, some of the chains I mentioned in my previous post are not among "the very very cheapest places to eat." Mea culpa...