I very rarely post on forums, but given the overall positivity of this thread (a breath of fresh air), figured I'd go for it. Keep in mind though that this is very much "favourites," not "best," and there are many RPGs I haven't played that have been mentioned on this thread. So, for what it's worth:
JRPG: Golden Sun 1/2
Golden Sun wasn't the first RPG I played, that 'honour' goes to 'Quest 64'. And by no means is it objectively the best either. The battle system isn't that complex compared to some other turn-based RPGs I've played, and better writing exists as well (not that GS has bad writing though - I like it more for its story rather than writing). But I include 1/2 because they were the first RPGs I was ever able to get into (excluding sub-genres). That, and where I feel GS shines is its overall plot and worldbuilding. There isn't some "big evil." There isn't anything insane that some JRPGs embrace, but none of the clichés found in WRPGs either (elves, dwarves, etc.). GS, especially in two, felt like a world that felt fantastic, but also down to earth. 1 had a good focused story, 2 had good exploration. And by the end of it...well, to be honest, I liked 'Dark Dawn', but I never felt the need for it.
Like I said, there are RPGs that are better than 'Golden Sun' out there. But this being a question of favourites rather than what's objectively better, GS takes the top spot as my favourite JRPG.
...though I'm still enjoying FFX. Remastered edition and all that.
WRPG: Mass Effect
Quick point, ME1 is the only 'Mass Effect' game I've played to completion. My Xbox 360 succumbed to the red ring of death at around the time ME2 came out, and by the time I got a new one, the trilogy had been completed. So yes, I intend to play ME2 and 3 at some point, but ME1 by itself is still a damn solid game. If 'Golden Sun' was the game that properly introduced me into the JRPG genre, then ME1 did the same for the WRPGs, and for many of the same reasons. Great characters, great writing, great worldbuilding. I know there are WRPGs that harken closer to the 'core' of what makes a WRPG a WRPG traditionally (e.g. 'The Elder Scrolls', 'Fallout 3', etc.), but while ME succeeds in worldbuilding, its characters, writing, and overall presentation is what puts it as my favourite WRPG.
TRPG: Fire Emblem: Blazing Sword
TRPG as in "tactical role-playing game." And like GS, I'll say this - there are other 'Fire Emblem' games that are objectively better in some areas (IMO at least). Storywise it's nothing special (bad guy wants to rule the world), and later games have added new mechanics to the series. But 'Blazing Sword' is the first FE game I played, and one of the few I've admittedly played at all. So in the area of favouites, this position still goes to BE. Had a blast playing it back in the day, loved the mechanics, story was enjoyable even if it was basic, and the characters were quite likeable as well.
ARPG: Diablo III
...Okay, I'll get some caveats out of the way. D3 introduced me to the 'Diablo' series in as much that it made me aware of its existence bar me playing some D2 with a friend in the early 2000s. As in, saw the trailer, I'd already invested myself in the 'StarCraft' and 'Warcraft' series, so I figured I might as well complete Blizzard's trinity. My inclination was to start with D3, but seeing the uproar that resulted when it was released prompted me to play D1 and D2 first so that whatever I could say about it, it could be said with some frame of reference. And of that frame of reference, D1 and D2 are definitely better in some areas. D1 has the best atmosphere IMO, and D2 probably exemplifies what many players would want out of an ARPG (more combat, less focus on story, etc.). However, D3 remains my favourite because of two reasons - combat, and writing. Combat, as in, the interface. As in, I no longer have to potion spam my way to victory unlike previous games, but can cast multiple abilities multiple times, and can adjust them based on the situation. It's not a "build," but I like the flexibility skill runes and abilities grant you, rather than the D2 method of locking a player into a certain build if they want to excel.
Writing...well, I'll say this. Every 'Diablo' game has a basic story. Of the RPGs mentioned above, even 'Blazing Sword' has a more intricate story than D3, in that D3 one could often guess where events were headed. But what I like about D3 is the writing. There's no longer long dumps of text. There's no longer long slogs in silence. The PC has a personality, the NPCs have a personality, the followers have personality, the log entries have personality, and I would often return to town after every quest just so I could hear what they were saying, how they'd get on with their own lives. I'd chuckle, I'd snicker, I'd be torn up when certain people died. D1 had a better atmosphere in of itself, but with D3, for the first time in the series, I felt like I was actually part of a world. I've played some ARPGs outside the 'Diablo' series, but I've come to realize that ARPGs aren't really my thing. D3 however, is the exception to that rule. Go figure.
(Though I do intend to try 'Dark Souls' at some point).
MMORPG: The Lord of the Rings Online
This is kind of a cheat, as LOTRO is the only MMORPG I played, and that was part of a course I took in game storytelling through Coursera, one I had to quit because of other academic commitments. Still, of what I did play of LOTRO, I thoroughly enjoyed. Combat is basic, 'story' is basic, but it's got what I suppose could be called the "WoW" factor. That for whatever reason, you just want to keep playing. I haven't gone back to it since due to how addictive it was, and because I don't think real-life could allow me to commit to an MMO, but hey, I had fun. Can't ask for more than that.