Going from earliest to more recent:
The Legend of Zelda: A Link To The Past - One of the first games I truly loved and was able to play over and over again in succession. It was a nice departure from games like Mario Bros. and Castlevania at the time, which I suck horribly at. I remember, I got stuck right before the Tower of Hera. I had to walk into the game store at the time in the mall near where I lived to find out how to do it, years after I had intially tried to play the game. And when I read how to do it, it was one of those d'uh moments for me.
Pokémon - When it first came out, it was awesome. But I didn't get my first copy until Gold and then the Yellow version was released, or whichever order they were released in. Then I had collected all the monster at the time, using a Gameshark for Celebi and Mew. That was when I lost interest, but until then I loved leveling up my Pikachu and my Charizard.
Final Fantasy VII - The first Final Fantasy I heard of and played. Since it was my first game in the series, despite not generally being a fan of the kind of setting, I loved it to death. I spent a whole month or two playing nothing but the game in my free time.
Final Fantasy IX - I got it the exact same time I got FFVIII, and ended up playing that one first because I thought I was going to like it a lot better. At the time, I didn't think I'd like the graphics for IX. Funny thing though, I ended up loving it and hating FFVIII. Beatrix was one of the first female characters I remember that wasn't the damsel in distress or 'girly' type, of which Tifa was more of. What Garnet went through with cutting her hair, paralleled something that happened to me at the time I was playing the game, so it was a very emotional scene for me. Rivaling that of Aeris' death scene.
Grandia - When exactly I got the game is blurry, I think I got it with FFVII but before VIII and IX. It was some game I never heard of at the time, so I thought it would be this obscure game I wouldn't like. I never thought that I would end up playing the game for a second time and spending 300+ hours in one playthrough. But I did, and loved every minute of it. I spent 180 hours on the first disk alone, most of the time was grinding to level up every character's skills.
It was worth it, made the majority of the game very easy. Though the side dungeons were still a bit on the challenging side, but much more manageable. And when characters left, they left behind skill books which contained experience toward the different weapon and magic types dependant upon what level they got those abilities to.
Final Fantasy IV - After I played this, it became my new absolute favorite Final Fantasy game, even though it was only the second game in the series I played. I liked it a lot better than VII, even if I couldn't beat the final boss without cheating. But I had to cheat to do the same for Safer Sephiroth in VII. Now, I don't have an absolute favorite because both FFIV and IX hold a special place for me.
Chrono Trigger - I didn't learn about this til Final Fantasy Chronicles was released, but when it was I did some research and couldn't wait to play it along with FFIV. The combo attacks were awesome and loved how Chrono wielded katana. And Ayla was my other favorite character with her fists. I played it non stop until I
Final Fantasy Origins (I and II)/Final Fantasy Anthology(V and VI) - I had fun with these, but only FFI and VI I played as much as I did FFIV and Chrono Trigger. In FFII, I exploited the system and started a new game after the first dungeon to exploit the level up system. I maxed out each character's Fist level because fighting with a character's barehands bypassed enemy defenses, made the Adamantortoise type enemies trivial along with the Trolls and Golems. The only enemy that the type of damage sucked with was the Emperor as he was programmed to really only take damage from the Blood Swords. So, one character I leveled up swords for the express purpose to dual wield the Blood Swords against the final boss. Oh, it was worth it.
Then FFI, I had actually downloaded an NES emulator and the game on my dad's computer before Origins came out to get a feel for the game. You could go through a few of the dungeons in any order you wanted, there was an item in a couple you needed to get to the next one, but you didn't have to beat the boss of the dungeon. So, I went to dungeons just so I could get to the third dungeon after them to level up quicker and get the class upgrades. It was worth it because the ice dungeon enemies posed no threat, except for the cockatrice with their Pretrify attack, and the volcano dungeon's damaging floors were also trivialized. Oh yeah, forgot to mention I used a party of nothing but Fighter characters, so because of the class upgrade they became Knights which could cast low level white magic spells. A set or two of 99 potions was more than enough to carry me as far as the class upgrade.
My experiences with these games led me to other games that I loved very much:
Legend of Dragon - The time I played it rivaled that of Grandia because I also spent time grinding to max out character's attacks and Dragoon Levels.
Grandia II
Grandia III
Grandia Xtreme
Dirge of Cerberus (grinded to get the Ultima Weapon gun)
Growlanser Generations (Growlanser 2 and 3)