I have a "real" computer that will run pretty much anything I care to try. I also have a little bitty computer with 1.1 GHz Celeron CPU, integrated graphics, 4 GB RAM, and running Linux Mint. I would like to have 1 or 2 games that I can run on the little computer for if I'm bored somewhere.

I've tried playing Bridge Constructor and Neverwinter Nights on it, but while they both seemed to run OK, the computer was getting uncomfortably hot to the touch.

I've looked at system requirements and they tend to specify things like "1.8 GHz" or "Dual Core Processor." These are pretty vague, especially since the number of GHz doesn't really mean that much these days.

Can anyone recommend any games that even my little computer will run without taxing itself and getting hot?

You can try games like Torchlight, Daikatana, Doom 1 & 2, Heretic and Hexen Collection, Fallout 2, King of Fighters 98, XIII, Deus Ex, Prince of Persia Sands of Time and The Warrior Within, Heroes of Might and Magic 3, Unreal and Unreal Tournament, Divine Divinity, Bad Mojo Redux, Downwell, VVVVV, Gato Roboto (probably), Eschalon 1 (it's free) 2 and 3, Postal Classic and Uncut (free too), Shadow Warrior Classic Complete (another free one), Dex (got a free demo to test), UnEpic, Butcher, and there you have to choose, but I recommend you to buy a base for your computer, it's always good.

Games that run in DOSbox or ScummVM should work fine. Most games from before 2001 should run well, which opens several genres. You can also look for ports for games like Quake.

You can't go wrong with Dizzy games.
https://www.yolkfolk.com/dizzyage/games.php?sortby=date&invert

I would just get a ebook reader and read some books.

Hmm - OP might struggle with 1995-1997 DOS Games on the Celeron 1.1GHz unless the Dosbox emulation has significantly improved since I had a 1 GHz AMD (I remember Syndicate Wars running like it was on a DX2-50). I'm not sure how much the improved IPC in processors will improve the experience (largely because I haven't had a processor clocked that slow in years).

Basically, I think OP should try it, but by buying and trying a single game first.

* Timespinner might work if you are looking for a metroidvania like the Igavanias. (On the other hand, I wouldn't try Bloodstained on that system.)
* How old is the system? If the graphics are Intel Iris or newer, Ikenfell is likely to work. If you have HD 4000 series (Haswell graphics) or earlier, it likely won't work. (It runs on my Celeron laptop via WINE, provided that wine-mono is available.)
* You could also try Nox Archaist, which is really an Apple 2 game. Apparently, the included emulator might not be so good on lower end systems, but other Apple 2 emulators should work. (I've actually been playing it on MAME on a Celeron, and even that's not the best emulator for lower end computers.)
* Of course, you could try old text based games. If you're into roguelikes, I hear nethack should be able to run well on that computer, as should Dungeon Crawl Stone Soup (I *think* that's the name); both those game should be available in the Linux Mint repositories.

Also, try turning on power-saving mode and seeing if the game still runs. This would force the CPU to run at a lower clock speed, reducing the amount of heat that's produced.

Edit: Also, you could try the games produced by Spiderweb Software. They're all turn-based, and they all have demos. (Note that Avernum 1 may have issues if you don't have a numeric keypad available; this was fortunately fixed in later games in the series and Avernum: Escape From the Pit.)

War Games :

"Unity of Command "
" Panzer Corps "

4x titles :

"Civilization IV"
"Galactic Civilizations II"
"Distant Worlds : Universe"
"Heroes of Might and Magic III"
"Age of Wonders II"

RTS :

"Command and Conquer 17 years anniversary package with community updates"
" Homeworld Remastered "
" Dawn of War II "

Rpg :

"Baldurs Gate Remastered"
"Skyrim"

at least... i would play those if suddenly my gpu crashed and i would be left with the integrated graphics chip

Creeper World 3

Go for FTL: Advanced Edition

It can run on a toaster and is a masterpiece.

Gorilla.bas

Doesn't get much more BASIC than that.

To an actual reply, Jeff Vogel's games have always been less about flair. They'll run.

I have plenty of GOG games running on my very low-spec 2GB RAM Celeron laptop. And they work like a charm.
I'd say that chances are good that every game before 2000 will run perfectly. The only issue is if you have a problem with old graphics quality.
My suggestion is to browse the GOG store and look for the games older than 2003.

EDIT:
Here are some suggestions from GOG's free games library:

https://www.gog.com/game/jill_of_the_jungle_the_complete_trilogy
https://www.gog.com/game/lure_of_the_temptress
https://www.gog.com/game/beneath_a_steel_sky
https://www.gog.com/game/teenagent
https://www.gog.com/game/postal_classic_and_uncut
https://www.gog.com/game/tyrian_2000
https://www.gog.com/game/treasure_adventure_game
https://www.gog.com/game/worlds_of_ultima_the_savage_empire
https://www.gog.com/game/ultima_worlds_of_adventure_2_martian_dreams
https://www.gog.com/game/shadow_warrior_complete
https://www.gog.com/game/eschalon_book_i
https://www.gog.com/game/stargunner
https://www.gog.com/game/flight_of_the_amazon_queen
https://www.gog.com/game/doomdarks_revenge
https://www.gog.com/game/bio_menace
https://www.gog.com/game/war_wind
https://www.gog.com/game/ultima_4

http://adarkroom.doublespeakgames.com/

Open-source text-based role-playing game, now with sound.

Since it runs on pratically anything these days... Doom

But seriously, what genre of games you like? That would make it easier to give suggestions.

Also, what Celeron gen is that? Depending on it, the integrated graphics can be more or less capable.

whats wrong with reading books?