I'm gonna be a complete bitch and say Ratchet and Clank cause it is Playstation only, but I grew up with that series :')

warcraft, darksiders, warhammer, disciples and some more :)

StarCraft and Brood War expansion

mass effect is number one for me, i really got sucked into this one.
witcher was also great, specially since i read all books before playing this game

1-Bioshocks 1,2,infinite 2-Half life 1,2,2 ep1,2 ep 2 3-Dishonored

TWD

It's a hard choice for me; Mask of the Betrayer, KotOR 2, Metro 2033, Deus Ex, Planescape: Torment, Fallout... But ultimately, I have to choose Star Wars: Republic Commando, for being the first game to make me feel like the NPCs who were fighting along side me had personality, far more than Bastila Shan from KotOR, Barney Calhoun from Half-Life 2, or any Jedi from the movies. I cared more at the ending than any time in, say, Final Fantasy 7, because I had reason to become attached; all of them were likeable, all of them were unique, especially because they were identical in terms of gameplay; it made their personality more pronounced than their abilities.

Planescape: Torment, Grim Fandango, To the Moon, many others

Fallout 1 & 2

Suikoden II or V. I'm biased towards Suikoden.

Too lazy too look through the page, but I'm seconding To The Moon and Planescape: Torment.

Mafia 1 - still one of the best cinematic experiences to date.

Prince of persia trilogy - as a whole it has pretty interesting story. individually - not so much.

legacy of kain series - one of the most complex, timetravel driven stories i have ever seen, heard or read.

gothic 1-2 - while the second game has more stuff to do, it has weaker story than the first game imo.

kotor 1-2 - glorious.

max payne 1 - great style, great characters. .. and great gameplay.

jade empire - underrated to say the least.

BRAID

If you take the time to actually read and understand the books, the story is outstanding

Planescape: Torment

legend among legends

Alan Wake, Portal, Metro 2033, Darksiders, Assassins Creed, Cave Story, Amnesia, Bioshock, Deadspace.

Max Payne 1-2
StarCraft + Brood War
MGS series
Deus Ex: HR

The Mass Effect series has one of my favorite stories.

The 仙剑奇侠传 and 轩辕剑 series.

Portal2 , Half-Life 1,2 ; Diablo 2

Alan Wake

Assassin's Creed series and Arcanum: Of Steamworks and Magick Obscura

Bastion

KINGDOM HEARTS!!!

Planescape:Torment - lore-wise and story-wise best game I ever played (bleh! It's best game I ever played at all ;P)