I'm a fan of the 3D Mario series, with Sunshine being my favorite.
I like Smash. Melee is the one I like the most.
I'm a fan of Mario Kart.
I like StarFox.
I love Rayman 2 and 3.
I guess you could say Madworld and Anarchy Reigns are a franchise? I like those. AR consumed my life for about two years straight.
I really like futuristic racers, my favorite games/franchise in this category being: Wipeout, Rollcage, F-Zero, Extreme-G, and Rush 2049.
I don't play many JRPGS. I do have some favorite franchises in the category.
The Grandia franchise has my preferred battle system. Grandia II is my personal favorite.
I got into the Atelier series starting with the first version of Rorona on the PS3. I hated it and Totori blew me away by comparison. Ever since Totori I've been hooked on the series.
Both Chrono Trigger and Chrono Cross are great to me. I like Chrono Cross more.
I like sports games. Both Basketball and NFL games are a yearly staple to me and my friends to this day. My personal favorites are:
NBA Live 95, 2004, 2010
NBA Street (I like the series as a whole, but love 2 the most)
NBA 2K series
Super High Impact
NFL 2K series (may it rest in peace)
I like actual racing games as well. Arcade-style preferred.
Need for Speed (Porsche, HP, HP2, U1, U2, MW, HP 2010). HP 2010 is my favorite. Once you get drifting down, the game is therapeutic.
Hydro Thunder (The whole Thunder series is good, but I like Hydro Thunder the most)
Daytona USA 1 and 2. We had a local arcade and that was the only two games I'd touch for the most part. I got torn away for Marvel from time to time...
When it comes to fighting games, I like them but suck at the inputs. Almost any fighting game I can watch and be entertained, but traditional-style 2D ones are too confusing to me. Unconventional fighters, however...
Smash series (already mentioned)
Power Stone series
Virtual Fighter series
Anarchy Reigns (already mentioned)
J-Stars Victory Vs (I hope the western release gets more people to try this game out. It's really good. Most people base their opinion of this game with videos on Youtube of people playing it that have no idea what they are doing and are making it look bad.)
That's all I can think of at the moment.