Anything with a lot of thinking involved, so turn-based strategy games or Grand Strategy games. Also some economic/ecologic simulations.

In the past, Panzer General 2/IIID, Fantasy General and Battle Isle 3 had that effect in spades. In fact, I was very sick one day and couldn't go to school as I was really in pain. I was so concentrated on Panzer General 2 later that day that during the game, my mind completely forgot about the sickness, even my nose stopped running for the time being.

These days, it's more the likes of Oxygen Not Included, Stellaris, Rise To Ruins, Hearts of Iron IV, though I also play those oldies from time to time for that effect.

Edit: And if I need to let off some steam, I'll put on some UT 2004

A Short Hike

2 hour super cozy game that makes me happy every single time.

Doom Eternal?

Seriously, I think right now it would be Hades. I'm at a point where I know the game well enough to win most of the time so I get to think up builds I want to try, use Keepsakes to get those builds, play through the game evaluating how I like what I devised.

I am a bit interested in Hades since everybody seems to love it. Perhaps I'll check it out myself.

If we are going with your definition of games you "play when you want to have it cozy? Maybe you have certain games that you can play over and over again, or that you have some sweet memories with." then several of the classic SNES games would qualify for me; Chrono Trigger, DKC 1-2-3, FFVI, FFIV, Breath of Fire I and II, Yoshi's Island. These are games I usually replay frequently (lets say, yearly) cause I just enjoy them, no matter how many times I've played them before.

Ogre Battle 64, Suikoden II, III and V would fit into this as well. Im currently replaying Suikoden II on my PS3.

Archibald's Adventures

Thief I&II fan missions

SSX3

Commandos

FlatOut 2

Colin McRae Dirt 2

Heroes of Might & Magic II

The Last of Us

Mirror's Edge

For many years my feel good game was the original Deus Ex but sadly I can't play it anymore.

I like SSX3 too a lot, I think it has a great soundtrack. Sad that you can't play Deus Ex anymore, that's a bummer. I have a version on the PS3, maybe you could go for that?

I might have to agree with the Civ series. Come back a couple times a year to lose myself in a campaign or two. That sense of discovery and planning in the early and midgame are hard to beat, though the late game can drag a bit.

Captain Toad Treasure Tracker!!!

I certainly need to play it again.