What do you call a game that has other games within it?

Solution 1:

TLDR: You wouldn't call it anything special.

The vernacular in regards to the overarching game is to use a possessive when referring to the minigame. So for instance, Final Fantasy's Chocobo Races. There is no special word or phrase given to the "main game" as you say.

The wiki entry for minigame does not mention a phrase or word to describe the "main game" either. IGN has a phrase that pertains to the minigame calling it "The Game Within The Game" but no phrase that describes the main game either. After further research I am unable to find anything suitable.

Perhaps something to look into would be: Gameification . I know that this doesn't have to do with the name of the overarching game, but it could perhaps lead you to some sort of answer.

This is a little different than including things like Chocobo Racing or Gwent inside of The Witcher series. Gameification basically means putting a gaming element around something that usually doesn't have those elements to increase user enjoyment or engagement.

Solution 2:

While other answers address the specifics of the example games given, there is actually a couple of different subtypes of games that would also apply to 'Game that has other games within it' that I think is worth covering:

Minigame Compilations

These are games where the entire goal/intent of the game is to overcome a wide variety of puzzles and challenges.

There may be little to no game mechanics outside of the puzzles/challenges. Plot/Story may non-existent, or only exist in a bare-minimum fashion to drive some sort of progression & unlock more or harder minigames/challenges.

While Wikipedia calls these 'Minigame Compilations', A large subset of games in this category would also be classified as 'Party Games', and purely singleplayer versions may also be referred to as 'Puzzle Games'.

Note that not all 'group-play' games are minigame compilations (eg. Mario Kart), nor does every Puzzle game feature multiple different minigames (eg. SpaceChem), so your partner may not be able to use 'Party Game' or 'Puzzle Game' for her academic paper.

Some examples include:

Wii Sports (Sport-based minigames)
Crash Bash (Party Game, loose single-player story)
Mario Party (Party Game)
Gizmos & Gadgets! - (Educational, Science-based minigames, loose singleplayer story)
1-2-Switch (Party Game)

See also: Wikipedia's List of Minigame Compilations

Game Packs & Collections

This classification is where the outer game acts as more of a 'launcher' for a collection of larger games (rather than minigames/challenges).

These types of games are generally called 'Packages', 'Packs', Compilations or 'Collections', the marketing folk among us would probably call it 'Product Bundling'.

Some examples include:

The Jackbox Party Pack - a collection of trivia-likes and other group/party games.
Rare Replay - a 30th anniversary collection of Rare video games (Rare the company, not as in common vs rare).
NES/SNES Mini/Classics - reproduction consoles loaded with classic Nintendo titles.
Mega Games 6 - A '6 in 1' compilation cartridge for the Sega Genesis/Mega Drive
Various 'Sega Mega Drive Genesis Collections/Packs', eg For PS2, and For Windows/Steam. - Because one can not own enough copies of Sega games.

See also: Wikipedia's List of Video Game Compilations.

The features of the 'outer game' can vary greatly, depending on the implementation. Most will have basic 'launcher' stuff like the ability to browse games and tweak some minor graphics and sound options, but other features can include:

Advanced graphics & sound - Scanlines & filters, colour/palette swapping, zoom/scaling, aspect ratio manipulation etc
Achievements and Challenge Modes
Progression/unlocks - eg. 'behind the scenes', concept art, cheats, mods/tweaks, bonus levels and even occasionally entire 'hidden' games.
Pause, Save state and 'rewind' functionality that may have not been possible on the original game.
Cheat engines and Mod integration

Conclusion

While larger games containing incidental minigames may not have an explicit name, there are games out there that almost purely focus on this idea, and those can be grouped and named: 'Minigame Compilation', or loosely 'Party'/'Puzzle' Games' would generally fit this scenario. While not quite what you're after, Game Compilation or Pack is also worth noting due to following a similar concept but with more rich games than just puzzles and challenges.