I have a couple of experiences that are memorable:
Xenosaga ~ This whole series was the most exceptional, thrilling experience I've had in the gaming world. I loved the story, the battles weren't all pushovers, and I've not played many games that made me think so deeply before. This series made me feel an array of emotions and that is a great feat in itself. On a smaller note, the main character was NOT a male that uses a sword! Wonderful.
Tekken 2 ~ I struggled with this game as a pre-teen so badly. I love and hate fighting games. I always get extremely tense when playing them and sometimes I have to pause mid-battle when I can't handle how close the match is or if I keep missing. That moment when I finally struck down Devil while playing as Jun, my favorite character, I couldn't even describe the feeling. A combonation of relief, amazement, excitedness, and satisfaction. Pure joy.
.Hack// ~ It started with a chance encounter at a gaming store at the mall. I knew I wanted a game, but I didn't know which game I wanted. I saw the cover of .Hack// Infection and thought it was pretty. So, I hid it in the store just in case I found something I wanted later, but surely enough I came back and bought it. Suffice it to say, after discovering .Hack I have loved almost everything it has to offer.
Star Ocean: Till the End of Time ~ I can't say how many hours I've played this game for. I loved this game so much. I didn't like the main character as much as I loved other characters, but the plot and gameplay were most enjoyable for me. I put it here mostly because this game had me enthralled for such a long time; most notably spending hours tearing through the ranks of the colleseum.
Mana Khemia: Alchemists of Al-Revis ~ The things I loved most about this game was the music and the battle system, as well as the graphics. It was cute, fun, and I enjoyed the entire soundtrack (which came with the game!). It wasn't as long as I wished it could be, but I'd play it more than once, most definitely.
There are more, somewhat smaller experiences with games such as Tales of Symphonia/Abyss/TheWorld, Naruto, and Final Fantasy, but then I'd just be going on and on. =P