Games with extraodinary good story [GA inside]

Firewatch :)

If you don't mind to get your brain working:

I really loved The Talos Principle (should also be on sale right now) and The Turing Test

ya fahrenheit have great story

and also u should give alan wake a shot

Planescape: Torment

Spec Ops: The line

OneShot (but don't read anything about it to avoid spoilers)
Mafia I & II (didn't play the third one)
Deus Ex series
Call of Juarez Gunslinger
Gone Home
Her Story
Dead Synchronicity
CHUCHEL (if you look for a good laugh and cuteness)
L.A. Noire

First Max Payne
Silent Hill 1-3
Some MGS(of course, if you have tolerance to special Kojima's vision)
Lots of Point and Clicks

I am not really sure about Quantic Dream. Have you considered the actual plot of Fahrenheit, for example? One of the protagonists gets hypnotised by a 3000-year-old Mayan priest who works for a shadow council to acquire a prophetic little girl so they can control the world. Said protagonist eventually gets killed and returns as a zombie, but despite being dead, manages to conceive a child with an unfertile woman. Additionally, we find out that the child is also sought by a sentient holographic AI that posed as an old woman in a wheelchair, because it also wanted to… do something unspecified. And to decide the fate of the child, the Mayan cabal manages to completely freeze the entire Earth until our zombie suddenly learns how to fly and do the kung-fu, so he can have a supernatural showdown on a rooftop where they fly around until one of them gets tired enough.

And David Cage maintains that this game is not a Lynch movie homage, it is supposed to be a very serious and REALISTIC crime drama.
(Disclaimer: The story, while playing the game, is actually dumber, but written words cannot express how much.)

Baldur's Gate (RPG)
Baldur's Gate 2 (RPG)
Mass Effect 1 (RPG / Shooter)
Portal 1 (has lots of puzzles though)
Warcraft 3 (RTS)
The Longest Journey 1 (Pt & Click)
Zelda: Ocarina of Time (N64 w/ Rumblepack)

Worth mentioning:
Eventide: Slavic Fable (HOG)

I recommend The Longest Journey saga. The first game is ooold, but the story is awesome and it gets you emotionally attached to the characters that are gonna reappear (or die) in the other two. The whole thing is beautifull :')

To the Moon
Rakuen
Finding Paradise

Divinity 2 and Divinity: OS2, Alpha Protocol and Spec Ops: The Line are my picks.

If you don't mind visual novels the Danganronpa games are really good.

I second Baldur 2 and Divinity OS 2

A lot of great choices already been mentioned by others (MGS series, Banner saga, Zelda, etc). But one I think you should look into is Uncharted series. AC series also had some good story telling.

Gabriel Knight 1-3
Oxenfree
The Cat Lady
Downfall
The Longest Journey (also maybe the sequels, although they are heavily inferior to the first installation in the trilogy)
Blade Runner
Unavowed
Gemini Rue
Primordia
Culpa Innata
a little bit on the "so bad that it's good" side: Black Dahlia

What remains of Edith Finch, is a game I didn't see in the above comments.

...are you looking for games that have a great stories or games that tell their stories well?

Here are a bunch of games that I really enjoyed the story in:

Planescape: Torment
Pillars of Eternity 1-2
Tyranny
Vampire: The Masquerade: Bloodlines
Vampire: The masquerade: redemption (not the greatest game, but the story is nice, in particular for those who know a bit about the lore)
Syberia 1 & 2
Wing Commander series
Mass Effect series
Torment: Tides of Numenera
The Longest Journey
StarCraft (I genuinely like this story, even if it's not a game that people usually mention for its story)
Baldur's Gate 2
Tex Murphy series (at least 3-5)
Ultima series, from 4-7 (these are a bit primitive, but 5 in particular really shows what a few moments of excellent writing can do for a story)
Gemini Rue
Valiant Hearts
I have no mouth, and I must scream
Alpha Protocol (for how much your choices actually matter more than anything else)
Avernum series
The Dig
Geneforge
Homeworld
Mafia 1 (2 is also quite alright, but not nearly as good, 3 is "meh")
Tales from the Borderlands
The Witcher series
Gabriel knight series
Shadow of Memories (also known as Shadow of Destiny)

• SOMA. Simply amazing
• Hollow Knight. Great lore, but the story is told in a very fragmented way, so not for everyone.
• If you dare with something really old, I think Future Wars: Adventures in time has one of the best science fiction stories in a point and click game.
• The Wolf Among Us
• The Return of the Obra Dinn. The story is good, but HOW is told, that's brilliant. Try not to spoil the experience seeing videos, trailers and let's play.

Brothers - A Tale of Two Sons
Gods Will Be Watching (contrary to what the Store page claims absolutely NOT a P&C Adventure game but rather a "Stress management" or Puzzle game)
The Blackwell Chronicles games

So many great games already listed. Two that need more attention:
Deus Ex - the original game
Planescape: Torment

Some extra games for the list;
Arcanum: Of Steamworks and Magick Obscura
Orwell: Keeping an Eye On You
Grim Fandango