Since Sticker Star just came out, I thought it’d be fun to discuss the series! I’ve loved these games ever since I was a kid. The stories are wonderful. ^^
Can’t believe I missed this thread! I love the Paper Mario series. I love all the Mario RPGs I’ve played, actually. It’s always surprising how invested you get in them…
I think The Thousand-Year Door probably is my favorite of the series because of the fantastically unique and memorable cast (seriously, probably my favorite cast in any video game), and the really fun and imaginative locations. The soundtrack is also top-notch but I could say that about every game in the series. The original is great too, although I feel like it’s a bit slow now. But I have a lot of fondness for it.
I feel like one of the few people that really did love Super Paper Mario. I can thoroughly understand why there’s a mixed fan reaction for it, but I think if you actually take your time to play it and immerse yourself in it, the story is unbelievably strong and there’s some very heartwarming moments. I enjoy the gameplay too, but I agree that it’s maybe not as fun as the others. Still, it gets major points from me for managing some serious tears outta me at times, something I wasn’t sure a Mario game would be capable of. (Ok, well apart from Mario Galaxy.)
I only just recently started Sticker Star and I’m quite enjoying it so far. The gameplay is fun and I like how much they’re implementing the paper theme. However, I am disappointed at the lack of unique characters and story elements so far, something the other three all had strongly. 
But I am only early on in the game and enjoying it for what it is a lot, I’m just bummed that Miyamoto is so heavily opposed to more deep plots in Mario games. I mean, c’mon, we enjoyed them a lot! These games were successful! Why would you try to remove that aspect of them? Ah well.