One that comes to mind is the game 'Legend of Legaia' for Playstation, a JRPG/Fighting game combination that I was obsessed with when I was a little girl. I completely idolized the female lead Noa and would often run around pretending to be her performing her attacks and saying what she would say in battle. Even to this day she ranks in my list of being one of the best lead characters in a video game.
The battle was also a lot of fun, the combat was turn based but you decided attacks based on a direction pad combination and you had to have enough 'Battle Points' to be able to perform special attacks. Then you had the ability to 'catch' certain monsters so you could use their magic in battle, it was a frustrating part of it... But fun none-the-less. Especially when you 'caught' a monster that gave you a lot of trouble and you get the joy of seeing it rip into it's former comrades.
The characters and story also remain one of the best in my head, maybe it had some cliches. But they were new to me and that was all that mattered. The silent hero Van, the strong but kind and naive Noa, the stoic and strong Vala... The side characters were amazing too. Several had great characterization, one side quest revolved around a pair of 'star crossed lovers' which still warms my heart to think about.
Er... Pardon the gushing. Guess I got carried away in the nostalgia.
I just wish I could find a copy so I could play it again.