A lot of this is dominated by nostalgia.
My life story.
1. FFX - My all around favorite Square game. Favorite soundtrack (tied with FFXIII), favorite story, and it's so much fun to play. My parents gave me this game for my fourteenth birthday along with a crapton of my favorite candy. I think this was my favorite video game related present ever. I didn't ask for it! They knew I liked video games and noticed I had two in the series already, FFVII and FFVIII, so they figured there were plenty of good choices. It was one of those rare moments where the non gamer knows exactly what the gamer wants. They weren't especially fond of this hobby of mine, so it was really nice. This was also a period in my life where video game time was precious, because I could only play on the weekends, and most of those weekends I was grounded from video games anyway! Any time I was playing a video game it was maaaagical.
2. FFXIII - This one just struck a cord with me. First of all, it was very pretty. Pretty music, pretty environments, pretty black magic, pretty pretty. It has my favorite cast, and it was my best experience going through a game for the first time. It wasn't quite as enjoyable the second time through, but that first run really packed a punch. I think it will always remain one of my favorites, though.
3. Threads of Fate - My first RPG and my first step away from games you could just play casually. I had to actually ask my dad if I could borrow the memory card he was using for Tomb Raider for this one! This was the first game I had any desire to "beat." Every other game I just played together with my younger brother or quit when it got past the mind numbingly easy part. (I would actually turn the Genesis off after I got past Emerald Hill Zone in Sonic 2.) It also showed me that games actually have pretty interesting stories, and it was more interesting than any other medium I experienced. Being ten years old, if it wasn't for kids, I could only really appreciate comedy. Dramas went over my head! But Threads of Fate was pretty kid-friendly while at the same time not being kiddy. I gave up on it after awhile though. I was stuck at the second dungeon and could not for the life of me figure out how to get out of it. I love Threads of Fate but that was the dumbest dungeon ever even if it did have nice music. So then I moved on to...
4. FFVIII - My first Final Fantasy! This one makes me feel the most nostalgic. The soundtrack is nowhere near my favorite, but every time I stumble across a theme from the game I immediately want to play it again. It's like smelling a pumpkin pie scented candle; you immediately have to have some kind of pastry. Triple Triad was oodles of fun, exploiting the Junction system was fun, and Jumbo Cactuar and his mustache. This one was quite an adventure trying to finish. The first time around I still didn't really know how to properly play an RPG, being only an hour into my very first one ever. I just spammed GFs at everything. So of course I got stuck at the missile base. Before I could make the connection that a machine would be weak to Quezacotl's electricity attack, my brother and his friend broke the PS1. The TV fell on it. The PS1 and FFVIII were smashed to a million pieces. When I was thirteen my brother got a PS2 and a friend gave me his copy of FFVIII, so that spring break I took another crack at it. I got stuck at the end of disc two then accidentally overwrote my file. Then I got stuck at the end of disc three and accidentally overwrote my file again. The fourth time through it was summer break and I finally figured out how to properly exploit the junction system. I also figured out that I was supposed to draw GFs from bosses. Finally it was finished, just in time for me to get my copy of FFX!
5. The World Ends With You - There really isn't another game like this one out there. I don't usually mention TWEWY as one of my favorites very often, but of all the games I've played, I've put the third most hours in it. Second if you don't count leaving the PS2 on overnight for the Neglamuur trick in FFXII. (First is FFX if you were curious.)
6. FFXII - I didn't care for the plot much my first time through, but it was enough fun playing it that I went through again. I liked it a little more the second time. I liked it even more the third time! Not many games grow on me like that with subsequent playthroughs.
7. FFX-2 - FFX-2 does not get the love it deserves. The writing...is not the best in the series, but it is so much better than people give it credit for. It was nice seeing how Spira just popped without Sin around. Also it is the funnest in the series.
8. Kingdom Hearts II - I just really like KHII.

And that's all. There are plenty of others I really enjoyed, but none that need a special mention.