WUT?! The story in FC5 is barelly above average (though thre are some colorful characters, thats for sure) and the ending is downright awful.

Among GOG games I would recommend The Longest Jourey and it's spin-offs/sequels Dreamfall and Dreamfall Chapters. However be prepared that in Chapters there is barely four descent puzzles for the whole game.

If you love reading, I really recommend reading the original Witcher books starting with The Last Wish before ever starting the games. The books (first two are short stories, then continued by novels building on them) are among the best fantasy literature I've ever read (and I've read tons...). The games are good in itself, but the emotional impact is a lot harder when you already know the characters and the world.

Sorry if I drop a spoiler or two,

I didn't say it was the best story ever, but it does, for me, Raise some questions, such as is it morally right to comitt acts of violence against another(or their beliefs) because you think your in the right? I'm talking on both sides of the issue, Father used violence to control, and you use violence to "end" the conflict.

and "wouldn't it be grand if you could just by petting, get a grizzly and a mountain lion as companions?"

For me the story even including the forced captures, proves that even the "good guys" can be the villians if they act on circumstancial evidence like we are forced to see in the begining.

The probelm is that while the game tries to raise those questions, it really does a poor job at that. Because it doesn't really show connections between the course and effect. You did what you did because the game made you do it. Was it right or wrong? No one can tell, because all the events in the end are not consequences of your actions but just game writers excersising their will.

I recommend Oxenfree, Blackwood Crossing and The Lion's Song. Those are more 'walking simulator" but I think they have a really good atmosphere/story.

Prince of Persia Sands of Time. Years afterwards, I actually took the time to tell that game's story to a friend, as if I was narrating a tale or summarizing a movies, and it was amusingly received with the same interest.

and

The Blackwell games. Not too fond of the last game, which goes a bit too overboard with its mystical denouement and all, but on a whole, great stories.

This! :)

I love this series, the final title indeed has some story problems but it's still enjoyable.

Do the Wizardry games actually have noteworthy stories? I always pegged them as pretty heavy (turn-based) hack and slash games, in which you spend the great majority of the time in combat ... (Haven't played them myself.)

Ultima VII is supposed to be great from a story-telling perspective, but the Ultima games are pretty old school, and not that easy to get into from today's perspective, IMO (I don't think the first games in the series have that much to offer in terms of good stories though).

Dragon Age would be a good choice, if you like the average fantasy story. The plot is not really all that special, but the story-telling is nice and there are some interesting characters in it (also a few different short stories as introduction depending on what class or race you choose; I created a character for each one and played through all of those beginnings before I decided which character to keep in order to continue the main story afterwards).

***

Anyway, since it hasn't been recommended yet, take a look at Night in the Woods. The gameplay is similar to Oxenfree but it's longer and personally I liked the story better (even though Oxenfree was interesting as well). It's like a cute graphic novel with serious themes (growing up, coping with loss and insecurity etc.).

And seeing as you're interested in Edith Finch, have you played:
Firewatch and
Tacoma?

Or, if you're into sci-fi/horror, you could check out Stasis
and SOMA (I haven't completed them, so I can't judge their stories, but they seem to be story-heavy games.)

(EDIT: Had to insert line breaks for the links because the forum is broken, sorry!)

Obvious +1 for Planescape: Torment. Could also back previous recommendations for Arcanum and, somewhat halfheartedly, maybe even Anachronox. But would add Betrayal at Krondor too.
And if I can add games that do the atmosphere very well, if not necessarily story, Vampire: The Masquerade - Bloodlines and Gothic 1 and, even more, 2 (didn't play 3 to know).

[Much later edit, +1 for Witcher games too, especially if played after reading books.]

I thought KOTOR had a cool story but I guess that hinges on you being a Star Wars fan or not.

You can try Planescape: Torment, Primordia, Sid Mier's Alpha Centauri and the Blackwell series, to name a few examples.

Not necessarily, but was part of the reason why I decided against mentioning those after mulling over it. Considered Deus Ex to, in which case it's more of a... conspiracy theories' fan? But none of these actually strike me as story-heavy.

The Witcher 3: Wild Hunt GOTY (Don't worry about starting with the 3rd)
Risen (Only this first one)
Shadowrun Series (Returns/Dragonfall/Hong Kong)
Saints Row IV: Game of the Century Edition (if you want some great comedy)
Fallout 3: Game of the Year Edition (I'm a heretic. I prefer FO3 to FO:NV)

That's why I wouldn't recommend this game series. The last game basically drops all plot threads promised by the previous one. Plus the ending more fitting for a fairy tale than urban fantasy. I was extremetly disappointed.

Wasn't Fallout 1, 2 and New Vegas more story-driven as compared to Fallout 3 and 4?