My 3 favourites, in no particular order, are:
FFVII - it was my first step into the world of RPG's.
FFVIII - as mentioned, the triple triad was seriously addictive. I also loved the look and feel of VIII.
FFX - The soundtrack of FFX is the thing that stuck with me the most.

The why's:

Characters:
All of them had characters you could relate to in some way or another. They were all special in their own way and allowed you a freedom of choice of which ones to use meaning your party felt like it was your party instead of being forced to use certain people. This left a lot of space for character and party levelling progression.

Minigames:
I spent countless days playing the minigames in all of the aforementioned games. Whether it be chocobo racing/raising, Triple Triad, or the wonderful Blitzball where you could even build your own Blitzball team from people you talk to scattered around the game world. They were all reasons enough on their own to play the game, and something to work on alongside the main storyline.

Storylines:
I was drawn into the story straight away for all 3 of these games. I felt like there was a sense of purpose and when you get drawn in enough you end up feeling like you can't log off because you need to do whatever the task at hand is in game otherwise X will die, or Y will destroy the world. The storylines just had to be seen through to the end. I quit playing FFIX half way through because I didn't feel it.