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One of the Fallouts has a text adventure game built in:

(I guess it;s more than one)

http://fallout.wikia.com/wiki/Pip-Boy_3000_Mark_IV_minigames

The outsidexbox folks did a video on this:

https://www.youtube.com/watch?v=AWTPgxsRAwc

Not sure what this is about, but I Dream of Jeannie has lots of genies in it, and it is fun. If you want something more game based, then off course there is the Shanteae series, which is here on gOg. But yes, lots of games have genies.

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In System Shock you could upgrade your firmware so you could play Space Invaders and a few other games.

In Anachronox you could play Breakout in a bar.

In Flight Unlimited there is an arcadecabinet in the office where you could play a fighting game.

GTA is loaded with them.

Rare used to do this a lot. Donkey Kong 64 had an arcade version of Donkey Kong you could play, and Goldeneye came with a secret ZX Spectrum emulator with a bunch of their earliest titles on it.

Celeste actually does this.

The thing is, Celeste was originally made for the fantasy console known as the pico-8; this game could be seen as a prototype, with only 30 screens, but it still featured the protagonist and her air dash. When the game was expanded into a full commercial release, a port of the original was included.

Of course, one of the best examples of games within games is the Nintendo DS title "Retro Game Challenge" (Game Center CX in Japan), which contains something like 8 games that you can play. The games are shorter than the retro games they're imitating, but are still a decent length (especially since one of them is an RPG). The game got a sequel in Japan, which I have played, and is quite good except for the fact that one of the games is a text adventure, which is not accessible if you don't know the language (I just found a guide online that simply gave a series of commands that would complete enough of the game to go on to the next one; playing a text adventure this way is not fun).

Edit: Since 227 mentioned some Nintendo 64 games, I believe the original (Gamecube) version of Animal Crossing has some NES games hidden in it; this feature is not present in any of the sequels, however.

Heroine's Quest the Herald of Ragnarok has gambling in it.

No love for Arcomage?

Super Turbo Turkey Puncher

http://doom.wikia.com/wiki/Super_Turbo_Turkey_Puncher_3

Stardew Valley has two arcade games you can play (accessed from two video game machines): Journey of the Prairie King (a shooter) and Junimo Kart (which is not like Mario Kart, but is a platformer where you have to jump a mine cart over gaps).

SWTOR has Pazaak, a card game.

Lost Viking in Starcraft 2.
Various mini games in The Witcher series.

If memory serves me well, in Marvel Ultimate Alliance you have to play a version of Pitfall using your superhero. Wish we could get that game here on GOG... and the sequel, which I never played because it wasn't available for PC back then (it was ported in recent years).

Star Wars KOTOR has some mini games like racing pods and gambling with cards.

In Sam and Max Hit the Road you can play Battleship. Also another mini game driving the car, but I never liked it so I don't remember what it was about.

EDIT:
Oh, I think you have to play Tic Tac Toe in Thimbleweed Park, can't remember if it was optional.

Always Sometimes Monsters has an arcade bldg with playable games in it