I could really go on all day about this subject.
The Legend Of Zelda, Links Awakening. (gameboy)
Probably the first game that showed me just how brilliant and involved games could be, it's also a stunning achievement when you consider the hardware it runs on. Has a genuinely sad ending too.
Chrono Trigger. (SNES)
I can't really say anything about this masterpiece that hasn't been said already. These characters are my friends, I feel like I know them and have had awesome adventures with them. I also just love the way the narrative starts off; you're just a guy (yeah, I name the main character after myself) going to the local fair and gets swept off on a fantastical adventure.
Panzer Dragoon Saga. (Saturn)
Team Andromeda built an incredible and unique world for their Panzer Dragoon series, everything had a sense of place in it and although totally alien, it was very believable. Panzer Saga really does feel like a Saga, and you can tell it was a labour of love, it almost has a fatalistic feel to it that must have mirrored the developers feelings towards Sega's downward spiral into near financial ruin. The game's a rollercoaster ride of emotion, especially for the time, but the ending is the real killer. It throws an amazing twist at you right at the very end, before leaving it open (not "shell out money for the next game to see what happens" open)
Shining series (Mega Drive - Saturn)
I just love the world and Character design that they produced for the whole series up until the point where Camalot got fed up with Sega's bullshit and went off to do their own thing (Golden Sun) Whilst Sega retained the IP and proceeded to spend the next 15 years raping it until it became unrecognisable. Amazingly, the fallout from all this was still being felt within the very small but dedicated community even recently with Japanese fans of the "new" games getting Youtube to take down videos of the older ones under the guise of copyright infringement.
Metroid Prime (Gamecube)
It's just an absolute fucking masterclass in game design. To be fair, so is Super Metroid on the SNES, but Retro studios deserve credit for taking the totally unique atmosphere of "super" and perfectly translating it to three dimensions and a new generation (there was an eight year gap between the games releases) If somebody were to ask me what I think the best game ever made is, this is my most likely answer.
God, I could go on all day, but that'll do for now.