Stuff my wishlist. Stuff it good. ( ͡° ͜ʖ ͡°) >>>>Update<<<<

Dark Souls
Torchlight 2
Silver (in GOG)
Transistor
The Witcher 1, 2 and/or 3
Borderlands 2 goty edition (or complete edition, even better)

Tomb Raider, Saints Row , Burnout, Mark of the Ninja, To the Moon, Dust An Elysian Tale, Rocket League ....

ARK Survival Evolved would be my recomandation, and Mass Effect series, since you have those exclusions

Bastion and Transistor for really beautifull soundtracks and hear touching stories.
Tormentum - Dark Sorrow for good artstyle (like on arts of Zdzislaw Beksinski and Hans Rudolf Giger)
Spec Ops: The Line for best story in shooters thus far
Psychonauts for... I don't know what to tell. In game you travel into minds of other people. And worlds inside their minds are different. For example, inside military mind always war, etc.
The Stanley Parable for... well, warch trailers, I can't give this game good description.
(Well, I'm not sure, that I gave good description to even on of games on this list)

As a COOP lover:
Killing Floor 2 / Don't starve together / Helldiver

Welcome to the site. here are my recom:

Geneforge Saga

Civ 5 better buy the complete...

Dungeon of the Endless

anyway, my wishlist is not long, you can check it out if you want. it will be easier if you gave us a genre you prefer

I would suggest:

Dishonored
http://store.steampowered.com/app/205100/

Witcher 3
http://store.steampowered.com/app/292030/

Banished
http://store.steampowered.com/app/242920/

Don't Starve
http://store.steampowered.com/app/219740/

The Escapists
http://store.steampowered.com/app/298630/

I know you don't want horror games, but i dont think it's really HORROR! :)
Darkest Dungeon
http://steamcommunity.com/app/262060

and almost forgot:
Sniper Elite 3
http://steamcommunity.com/app/238090

and if you are in for a good old rpg games, i recomend Fallout 2. I even cant count how much time i played it in childhood!

Some great titles from last year: Rocket League, Cities Skylines, Hand of Fate, Prison Architect.
Also, if you like open-world games then try some titles from Avalanche Studios: the Just Cause series, Mad Max, Middle-earth:Shadow of Mordor GOTY.

Far Cry: blood dragon
Hotline Miami
Hard Reset
Saints Row

And for something completely different:
Cinders
Gone Home

Also i would recommend:
Dwarfs!? Not everybody might like it, but there is f2p version of it you can try (no microtransaction s*** - just plain game cut to normal difficulty).

Also wanted to mention S.T.A.L.K.E.R. but it might be considered as horror game :)

I'd say Dishonored and Bioshock series

I suggest Xcom 2 as it looks like they are going to fix all the problems with the first game, plus the modding potential, oh my.

Hard Reset
Crypt of the Necrodancer
Ori and the Blind Forest

Divinity: Original Sin, Dragon Age: Origins, Deus Ex: Human Revolution, Shadow Warrior (the new one).

Rally sim ; Dirt Rally (really hard but amazing !)
Survival : The Long Dark (great ambience and the campaign will be soon available)
Rpg : Pillar of Eternity
If you like Subnautica, you might want to check Stranded Deep
And if you want small game i can recommand ; Binding of Isaac Rebirth, Hand of Fate, Knights of Pen and Paper +1,
Papers Please, Hotline Miami, Choice of Robots, The Stanley Parable, ...
And so many more ! (Witcher 3, Rocket League, Crusader Kings 2, Killing Floor 2, Chivalry, ...)

How about some strategy games? I personally recommend any game by Paradox, especially Europa Universalis IV and Crusader Kings II. Although you probably know them, in short they allow you to control any country or territory (in CK2) and manage virtually every aspect of it, including technology, commerce, politics and diplomacy. They are generally very complex, but also incredibly rewarding and addictive once you get the hang of it.

bastion
http://store.steampowered.com/app/107100/

trine 1 & 2 ?

Suggesting Binding of Isaac and Rogue Legacy if you like rougelikes, dying Light if you like zombies, brothers: a take of two sons if you like to have feels :D

Any Telltale game ever. Tales from Borderlands for example.

Transistor - Beautiful art and soundtrack, and I found the combat really fun.
Psychonauts - A pretty fun game with a decent sense of humor.
Luftrausers - A fun, kinda bullet hell-y game with quick rounds.
Downwell - Another game with quick rounds and an example of a simple concept done well.
Receiver - Not the most polished of games, but it's an interesting concept. I've had a lot of fun with this one.
Recettear: An Item Shop's Tale - Play as the item shop owner in an RPG. Capitalism ho!

Fallout 3 and Fallout: New Vegas, along with the ones that aren't FPS.

Hmmm... Maybe 60 Seconds?

If you've played the witcher maybe you can try dragon age? Or the bioshock series? If you're into rpg/story mode then I definitely recommend to the moon. Not sure if the walking dead game is "horror" but it's also good. Tomb raider is also a great experience

PIXEL PIRACY :D

If you like Assassin's Creed style games, I recommend you Middle Earth: Shadow of Mordor.
By myself I recommend you Mark of Ninja, because it's my favourite game I ever played. Awesome 2D-stealth. Recommend to buy with Special Edition DLC, because it's add more features and extra levels, that really good, because only minus of base game - it haven't extra levels or map editor. And Left4Dead2 you can play it like forever because of AI Director system, Steam Workshop and a lot of modes and user created maps (look for them on sites like fpsbanana).

Just a few that I have spent (too much) time on:

The Journey Down (2 parts published)
http://store.steampowered.com/app/220090/

Deus Ex Human Revolution
http://store.steampowered.com/app/238010/
(to prepare for the next one...)

Fallout New Vegas
http://store.steampowered.com/app/22380/
(to keep one busy until FO4 is cheaper)

Terraria
http://store.steampowered.com/app/105600/

Psychonauts
http://store.steampowered.com/app/3830