The List

The full list of games with a "story" that I've completed.  I like to include this as it seems to be the best conversation starter for those who have actually played the same games, and it also will give a gauge as to weather or not you might have the same interests as I do.

Currently Playing - Destiny, Diablo III, Telltale's Batman, NBA2K16

Most Recently Completed - Battlefield 1

Playstation 4-
Assassin's Creed IV: Black Flag - 2/24/14
Killzone: Shadow Fall - 3/10/14
Infamous: 2nd Son - 7/19/14
Injustice: Gods Among Us - 7/19/14
The Last of Us Remastered - 9/1/14
Destiny - 10/6/14
Assassin's Creed Unity - 12/20/14
Shadow of Mordor - 2/2/15
Journey - 8/17/15
Uncharted: Drake's Fortune - 10/18/15
Gone Home - 1/13/16
The Witcher 3:  Wild Hunt - 2/11/16
Borderlands 2 (The Handsome Collection) - 3/6/16
Minecraft Story Mode - 4/3/16
Shovel Knight - 4/28/16
Uncharted 2 - 6/9/16
The Division - 6/27/16
Destiny: The Taken King - 9/5/16
Wolfenstein: The New Order - 10/17/16
Tales From the Borderlands - 11/5/16
Borderlands 2: Captain Scarlett DLC - 11/6/16
SOMA - 11/20/16
Battlefield 1 - 12/4/16

Playstation Vita-
Uncharted: Golden Abyss - 2/28/15
Persona 4 Golden - 9/13/15
Hotline Miami - 11/7/15
Gravity Rush - 1/19/16
Killzone: Mercenary - 3/7/16
Sound Shapes - 4/27/16

Wii U-
Game & Wario - 3/18/14
The Legend of Zelda: The Wind Waker - 4/21/14
Super Mario 3D World - 2/16/15

Xbox 360-
Ghost Recon Advanced Warfighter
Gears of War
Assassin's Creed
LEGO Star Wars II - The Original Trilogy
Marvel Ultimate Alliance - 2/17/08
Beautiful Katamari - 3/29/08
Bioshock - 7/6/08
Portal - 9/20/08
Braid - 3/7/09
Grand Theft Auto 4 - 5/17/09
Mass Effect - 12/18/09
Fable II - 2/15/10
Final Fantasy XIII - 2/20/11
LEGO Star Wars - The Complete Saga - 4/19/11
Halo 3 - 6/13/11
Ilomilo - 6/14/11
Half Life 2 - 8/21/11
The Elder Scrolls IV - Oblivion - 10/19/11
Portal 2 - 11/8/11
Plants vs. Zombies - 11/25/11
Borderlands - 11/28/11
Sonic Generations - 12/24/11
Halo 3 ODST - 1/1/12
The Elder Scrolls V - Skyrim - 1/19/12
Half Life 2 - Episode One - 1/29/12
Rayman Origins - 4/12/12
F.E.A.R. 3 - 5/25/12
Halo: Reach - 7/20/12
Hunted - The Demon's Forge - 8/10/12
Half Life 2 - Episode Two - 11/11/12
LOTR - War in the North - 12/17/12
The Walking Dead Season One - 2/8/13
Bioshock Infinite - 4/16/13
Halo 4 - 8/26/13
Fallout 3 - 11/4/13
Diablo III - 1/13/14
Tomb Raider - 4/29/14
The Wolf Among Us
The Walking Dead Season Two - 8/31/14

Nintendo 3DS-
Super Mario 3D Land - 2/6/12
New Super Mario Bros 2 - 9/4/12
Legend of Zelda: Ocarina of Time - 1/17/13
Kid Icarus: Uprising 9/8/13
Luigi's Haunted Mansion: Dark Moon 10/21/13
The Legend of Zelda: Four Swords - 3/19/14

Wii-
New Super Mario Bros Wii - 2/4/13 
Kirby's Return to Dreamland - 2/24/13 
Super Paper Mario - 8/26/13 


Nintendo DS-
Civilization Revolution - 5/3/10
New Super Mario Bros - 7/29/10
Hotel Dusk, Room 215 - 9/14/10
Final Fantasy III - 4/18/11
Mario and Luigi - Bowser’s Inside Story - 10/25/11
Mario vs. Donkey Kong - Mini-Land Mayhem! - 3/2/12
Chrono Trigger - 7/6/12

Game Boy Advance-
Super Mario World (Super Mario Advance 2) - 5/9/09
Metroid Fusion - 7/10/09
Final Fantasy 1 - 10/11/09
Advance Wars 2 - 1/19/10
Wario Ware, Inc. - 2/2/10
The Legend of Zelda – The Minish Cap - 3/18/10
Final Fantasy II - 6/29/10
Wario Ware, Twisted - 12/14/10
Super Mario Bros. 3 (Super Mario Advance 4) - 1/18/11
Yoshi’s Island (Super Mario Advance 3) - 1/14/11
Super Mario Bros 2 (Super Mario Advance) - 6/13/11

Playstation 2-
Grand Theft Auto III
Grand Theft Auto – Vice City
Grand Theft Auto – San Andreas
Dark Cloud
Katamari Damacy
Zone of the Enders
Shadow of the Colossus
Prince of Persia – The Sands of Time
Metal Gear Solid 2 – The Sons of the Patriots
Balder’s Gate 2 – Dark Alliance
The Lord of the Rings – Fellowship of the Ring
The Lord of the Rings – The Two Towers
Resident Evil 4
SOCOM: U.S. Navy Seals
Star Wars - Battlefront
Shadow of Destiny
Metal of Honor – Frontline
Twisted Metal – Black
Time Splitters

Xbox-
Halo
Halo 2 - 2/16/08
Star Wars - Knights of the Old Republic - 9/18/08

Playstation-
Final Fantasy VII
Tomb Raider 2Resident Evil 2
Driver
Driver 2
Metal Gear Solid

Nintendo Entertainment System-
Super Mario Bros
Mike Tyson's Punch Out

PC-
Transylvania
Kings Quest
Police Quest
Return to Castle Wolfenstien
Leisure Suit Larry

I also wanted to include a short list of the games that essentially have no story, but that I've put enough time into to warrant mention...
Playstation 4-
Resogun

Wii U-
Mario Kart 8

Xbox 360-
Forza 2
Pinball Hall of Fame - The Williams Collection
Carrcasonne

RISK Factions 
Catan

Nintendo 3DS-
Mario Kart 7
Mario Golf World Tour

Playstation 2-
Final Fantasy XI
Gran Turismo 3
Tekken Tag Tournament

Playstation-
Gran Turismo
Gran Turismo 2
Tekken 3
Star Wars – Masters of Teras Kasi
Street Fighter Alpha 3

Nintendo Entertainment System-
Duck Hunt
Ice Hockey
Tecmo Bowl 
 
PC-
Dr. J vs. Larry Bird  

....and believe it or not I also enjoy sports games.  I do not run out and buy the latest version on day one, however I've played so many different Madden, Tiger Woods, NHL, MLB, and NBA games that I won't bother listing each version.