Alright let's do this. Note off the bat: a lot of the national museums are overrated while the city-owned museums are the opposite, and free. Also I'm limiting it to MUSEUMS, not sights. So Catacombs aren't included but the Crypte d'Archeologique is. Also note the 1st Sunday of every month all of the national museums are free.

Paris museums that I've been to* ranked:

1. D'Orsay. They turned the old train station into a ludicrous collection of this very specific type of art. Gets so much traffic
2. Picasso. I'm a HUGE Picasso fan.
3. Petit Palais--as much for the building as the art.
4. Carnavalet (do this your first day. It's free and gives you a great baseline for Paris)
5. Arts et Metiers. It's more of a museum of inventions than art. Absolutely fascinating from prehistoric tinker toys to computers of my childhood.
6. Rodin on a nice day. Don't bother if you can't sit outside for a bit. Secretly awesome place to bring a Kindle and chill.
7. Crypte Archeologique. Fight me (everyone who travels with me does). Absolutely incredible to literally walk the streets of Roman Paris underneath the square in front of Notre Dame, then see the 3D maps.
8. Art and History of Judaism. No this is not bias--it's the bet of its kind in the world even if you don't "get" all of it. This museum is what made my wife and me fall in love with Chagall.
9. Navy (National de la Marine). Ironically because the French were so bad at Navy and so good at Army their Army museum is huge and overly nationalist while their Navy Museum is fascinating and...like...honest about shit. Lots of model ships and paintings and a full sail you can touch and all sorts of other bits from the Age of Sail. It's right by the Eiffel Tower so when you're kind of underwhelmed by that you can go dad the fuck out in here.
10. Orangerie. Cute little museum in the middle of the Jardin du Tuilleries that has a lot of what you might want to get from the Louvre or D'Orsay without spending all day at a museum. Depending on exhibition this might be in the Top 5.
11. Louvre: Sully. The Louvre is really 3 museums and my favorite is this one, which starts you at the base of the old medieval castle (that's what the Louvre used to be) and then the antiquities. Venus de Milo is here. It circles the Court Carree, which is where I got engaged. Oldest part of the building FYI.
12. MAM (Modern Art Museum) (no not THAT Pompidou). Like Carnavalet it's owned by the city so it's free and they have a great collection that's actually geared to the public.
13. Montmartre. This was Renoir's house and I unfortunately was leaving Paris after studying abroad the day before it opened. I went back in it's an amazing find, with a medieval vineyard in the back.
14. Armee (aka Invalides). Aka the Dad Museum. France is super proud of their landed military history, which means they have a great collection but it's not put together well because it's so infused with Napoleonic nationalism. Needs more 100 Years War and not go so hard on the Austerlitz. 
15. Louvre: Richelieu. When you go by the Louvre you wonder how to get into the sculpture area..it's this wing. There's also a recreation of a Persian temple, the Medicis Gallery, and everything else that didn't fit into the other two wings, which makes it a better museum IMO because it's eclectic. It's also the least crowded wing.
16. Mundolingua. Accidentally found this gem between my favorite place in Paris (Jardin du Luxembourg) and going to see Saint Sulpice. It's interactive and kickstarted my interest in linguistics.
17. Cluny (Moyen Age)--I love it, and that you're in a roman bathhouse. But medieval European art is actually pretty uninteresting.
18. Decorative Arts. Well-done museum that actually made me interested in the history of toys and clothing.
19. Plans-Reliefs (Museum of Maps). Dad museum for sure, but it's all the King's maps and a lot of battles in 3D from Napoleon or WWII and a map of New York circa 1796 that you're like "Why do they have this?" and then go "Oh shit they were probably planning an invasion of the U.S. during the Quasi-War"
20. Cognacq. No it's not about wine. It's a private collection of Rembrandts and Cezannes and Degas that was donated to the city. Takes you 10 minutes and it's free.
21. Architecture: Went with an architect which enhanced the experience but the old stuff is pretty cool. Kinda small for how much they charge.
22. Sewers and the Pavilion of Water. Two museums ppl usually see together. Okay don't laugh but it's actually a pretty good museum if you can take the part where it smells. I had to see it because I was working for Pollution Engineering magazine when I went, but it's actually pretty fascinating to go through the history of shit.
23. Louvre: Denon. No I don't actually rank the Mona Lisa and a lot of other great works by Italian artists behind the history of shit, but they were close enough that I committed to the bit.
24. Natural History. Redone recently so might move up next time. It's on all my visit recommendations though because it's in the Jardin des Plantes and for some reason tourist guides never tell you to investigate this area, but you can get an amazing day out of this, the garden, the grand mosque, Rue Cuvier, and then up to the Mouffetard, which is like the State Street neighborhood for La Sorbonne but with great and less expensive dinner (if you can take walking uphill).
25. Opera. This is only very technically a museum; really you're there to see the Opera House for free.
26. Air. It's the environmental museum but it's also the top of a terraced hillside garden up in Belleville where you can get an AMAZING view of the City of Paris.
27. Wine. Free. Free wine. Wine bottles from the 1400s...shit I had you at "Free wine."
28. Science and Industry. Not something American tourists bother with but it's a really good one. Went for a huge Star Wars exhibition.
29. National Archives. Better than the US one because it goes back to the 7th century, but there's only so much you can ooh at paper.
30. Gobelins. If you're into tapestries, this is where they were made. But after 5 minutes someone you're with goes "so, they're carpets?" and the spell is broken.
31. Marie Curie's Lab.
32. Grevin. Not the best wax museum, honestly. More for the French than American tourists. But I went when The Body was making its tour.
33. Cinema. The topic is very interesting--the French drove cinema history for 100 years and they know it, but they didn't have a very good way to present it when I saw the museum. Might have changed things up since?
34. Bible and Holy Land. It's only open on Saturday, which screw you Archdiocese of Paris: you take our stuff and only make it available on the Sabbath? J/K it's free inside the city's Catholic Institute so if you're by there on a Saturday you can pop in and see some extremely old artifacts.
35. Museum of Magic. We've reached the "Paris has to have a museum for EVERYTHING" line here.
36. Museum of Music. Thought it would be cooler but it's trying too hard to be like an American museum of rock n roll instead.
37. Dali. Meh.
38. Pompidou. I find it ugly and built more for entertaining super fancy rich functions than appreciating modern art.
39. Camondo. I'm not that into furniture so maybe it's different for you.
40. Police Museum. It's free and actually done well, just I was less interested than I thought I would be in "the gun that shot a guy I've never heard of." Needs to go deeper into history--right now it's mostly stuff the city pulled out of its evidence lockers and the rare guillotine that wasn't destroyed.
41. Louis Vuitton. Standard overpriced museum that's actually more meant for functions than public viewing.
42. La Poste (the mail). Paris has a museum for everything.
43. Counterfeiting. EVERYTHING.
44. Luxembourg. Skip it. Expensive gallery to hold events for politicians that's filled with mostly British (?) paintings for some reason.
45. Hunting and Nature. Just some dude who liked hunting's collection. I don't really ooh and ah at animatronic talking boars or Napoleon's guns.

Not ranked: Grand Palais and Jeu de Paume. You have to check what exhibit's there. The Shoah (Holocaust Museum) is like any other outside of Jerusalem's unless like me you have some relatives' names in there. If you've been to a Holocaust Museum before you don't have to do one on your Paris trip.

*I haven't visited Quai Branly but people who have say it's spectacular. There are a few more I've visited that I can't remember...you walk around Paris and dive into a museum of Oriental Art or Edith Piaf or something and go "That's nice."