Mario & Luigi:
Dream Team by far. No doubt the best music, the longest and most challenging game in the series, and most replayability.
Paper Mario:
Well it would be Super, but thats more of an adventure/platformer, so I guess Thousand-Year Door takes it. No real explanation needed.
Pokémon:
This is challenging because every generation had something. 3 had lolstalgia, 4 had Battle Revolution and I had crazy fun with the multiplayer on that with friends (not Colosseum or XD because we didn't try to make competitive teams in gen 3), 5 had online multiplayer that finally worked well, and 6 has the best music/visuals. I guess I'll go with Black/White because Ubova was my favorite region, it was so diverse. I lost Black 2 shortly after buying it but that game seemed better. Maybe I should rebuy it eventually. :/
Final Fantasy:
I'm gonna cheat here and say Bravely Default (should be getting Second next month and I have a good feeling that that one will be better). Fun/funny classes, amAZING music, and every character felt so detailed. Also Victor and Victoria are the new age Jesse and James.
Fire Emblem:
I seriously can't choose, I'm gonna avoid this one. (Especially because you'll be criticized no matter what game you pick by the "true fans" lmao you aren't a fan if you hate every game). I will say that I've played all of the English games besides Radiant Dawn and Shadow Dragon and I started before Awakening released so I'm not an Awakening bandwagoner. (With Sacred Stones specifically).
Mother/Earthbound:
3!!!!! LUCAS IS MY BABY AND HIS GAME MAKES ME CRI EVRYTIEM!
...I'm so sorry, I promise that I'll never do that again.
Xenoblade:
Well I've only played X so...
I feel like I've played from more series than just this but I can't remember them. Oops.