1.
Osmos (bought on Humble Store. You're a cell that wanders around eating smaller cells to grow bigger)
https://youtu.be/jrzhlTn1_ds

2.
Complements (freeware game where you clear the board using complementary colors)
https://youtu.be/Dkw8xOtFSio

3.
Fractal (puzzle game, bought on Humble Store)
https://youtu.be/DQJnPEJ6HtQ

4.
Fear (bought here at gog)
https://www.gog.com/game/fear_platinum

Fear 2 (bought here at gog)
https://www.gog.com/game/fear_2_project_origin_reborn

(I've played them so much that I can get "in the zone" and relax while playing)

5.
Gluey 2 (flash based browser game with physics)
https://youtu.be/8i_v8sOmVNY

Chess against a very low AI. It also serves as a good measure of how sleepy you are.

Hidden object games. It may be their main function.

The cube escape rusty lake series of adventure games.

Freecell quest. For extreme cases.

Reigns.

I resisted. You did not!

Happens to me more than I would like to admit; being so tired I barely have the desire to game. It is really interesting how much it depends on how rested I feel. But if it gets critical, I have no problem getting 2 hours of sleep after I get back from work. Better than spending the rest of the day in zombie mode.

But when I decide to power through it, I don't feel like starting anything requiring much thought or commitment. So I usually just end up playing bot matches in a randomly picked Unreal Tournament. Bound to keep you on your toes and time flies by so fast it's incredible.

I don't usually play while being tired, but when i just want to have some calm gaming fun i usually go for Microsoft Flight Simulator 2004: i enjoy getting onboard some GA aircraft and just casually fly around over mountainous regions, because i enjoy sightseeing (FS2004 is beautiful with the right addons); LOWI is my go-to airport most of the time. (https://innsbruck-airport.panomax.com/?cu=en-GB)

Ouch. :D

Anything that releases adrenaline to your veins.

For me the best way to not fall asleep is to play Team Fortress 2. It is a competitive online action game, so yeah, it usually pumps me full of adrenaline and I am usually fully awake.

For single-player games, just something that keeps your attention well. For me e.g. a good realtime strategy game can do that, I am so focused on it that I couldn't think of sleeping.

Staring at LED screens in itself tends to keep you awake, thanks to the HEV light, also known as "blue light". Unfortunately it might also damage your eyes in the long run (speeding up macular degeneration). I took the question more like as "I have to stay awake for reason X. Which game would keep me awake?".

First off, reduce your monitor brightness. Keep it preferably below 30.

If you still feel strain in your eyes when looking at the display, try downloading Flux or similar software. You can use it to reduce the amount of the harmful blue light. It destroys color accuracy but you get used to it after a while. I don't use it when gaming but at work, where I stare into a display almost all the time, I have it between 5000-5500K depending on the surrounding light at the moment (the default is 6500 Kelvin, so about 20% reduction) and it made a huge difference in my ability to stare into a display for prolonged periods of time since I started using it.

I have actually done that. At my work, I have used the monitor settings to reduce the blue color (red and green at normal). It does change colors a bit like white getting a slight yellow or pink tint, but it doesn't matter especially with desktop work. It might matter if I was some graphics designer using Mac, but luckily I am not.

I also recall it was possible to achieve straight from Windows without any extra utilities (and probably from Linux), but I have the instructions hidden somewhere. This in case you are not using an external monitor with its own color settings, but e.g. a laptop which doesn't have any extra monitor controls, besides mere brightness.

Depends. If you want to stay awake (since you describe not wanting to go to sleep during the day):

Amnesia - The Dark Descent
F.E.A.R.
Overload

Or the good old Doom and Quake games. Will get your adrenaline running and chase away the tiredness.

But if you want something mellow that can be played without much mental presence, go with Visual Novels as kbnrylaec suggested. There are lots of those now on GOG. Or walking simulators. Try Dear Esther, that has a dream-like atmosphere that meshes well with being half asleep.

Astroneer - https://astroneer.space/ - a very peaceful and relaxing game that is interesting to play but never get stressful. It is a little bit of a slow slog and grinding, but when tired that's what I am looking for anyway.

Torchlight II or any other aRPG. No cognitive skills needed, and the regular jolts of reward and progress keeps you awake.

Like someone else mentioned, bot matches in any game.

Any Civilization game. You will forget that you're tired, hungry and in need of a bathroom or a shower.

Rimworld, because you can load up a few orders and then sit back and nod off while your colonists begin to follow your commands. It's also fun because a few minutes later all hell will break loose, yet you'll be too tired to do anything about it. And watching a colony unravel and completely go to shits in Rimworld is part of the fun.