Gwent!

All of the Yakuza games are basically, collections of well made mini games that turn each beat-em-up campaign into a hundred hours of fun. But among those, the Cabaret Club and Pocket Circuit RC race-car games from Yakuza 0 and Yakuza Kiwami, are probably my favs.

Final Fantasy VIII card game ♥️

Animal Crossing on the GameCube straight up let you buy little NES consoles with a small variety of titles.

I have a few.

PGR3, a Xbox360 racing game, contains Geometry Wars 1 and 2 as mini games. YT Link

Celeste contains the entirety of Celeste Classic (PICO-8) as an easter egg in one of its levels. YT Link

Xenogears, a PS1 JRPG game, contains a battle arena minigame, and I spent a few hours on that as a kid. YT Link

Machinarium's Gomoku/5 in a row minigame is so much fun, I played it with my friends at school when we didn't want to listen to our teacher :) By the way, I really recommend Machinarium to every fan of old school point-and-click games.

Gwent from Witcher 3

I remember spending so much time playing Farkle in Kingdom Come Deliverance, betting my money on every game. I think Witcher 1 or 2 have similar dice game that i also very into it, played with every NPC possible whenever i have the chance.

FFX Blitzball is the mini-game that I sunk the most time into by far (100+ hours), and always had fun.

Gwent from Witcher 3 kind of goes without saying, the framework is so good it's spawned 3 full games that I can think of.

Best Hacking mini-game goes to the newer Deux Ex games, quick, the right amount of challenge but if you didn't like it you could basically never do it.

Best lockpicking I'm going to give to Starfield. Literally the only part of the game I actually enjoyed, each is a great little puzzle.

As a fan of the LucasArts point-and-click adventure games of the 80s-90s, it would be remiss of me not to mention that Day of the Tentacle, the sequel to Maniac Mansion (their first adventure game ever), actually contains Maniac Mansion as a minigame.

Kinda old school here but I really loved Pokémon Stadium mini games, also the shooter mode from Donkey Kong 64 was a blast, back in the day. Even Banjo Kazzoie/Tooie had some amazing mini games, I really loved those.

In Super Pitfall for the SNES, there was an Easter egg hidden in a temple that would warp you into the original Atari version. If that counts, that's my favorite "game inside a game."

Gwent, in The Witcher 3!

The host club manager minigame in Yakuza 0 and Kiwami 2, I don't even remember their rewards, just that they were very fun. Compared to the real state minigame in 0 that was so boring I only remember the prize at the end, at least it was worth it

Rdr2 fishing has a following…

Blitzball

The door hacking in Deus Ex Human Revolution. Each one was unique, could be solved by skill (speed and precision) or with tools (consumable items found throughout the game). It was a mini puzzle game each time you tried to unlock something.

At the time, I loved it so much I tried to build my own version but it never went anywhere.

I really liked the hacking puzzles in Half-Life Alyx. There was a nice variety to the different type of puzzles that could appear, and the difficulty never felt like it got out of hand.

I spent way more time than I should have playing Dice in Kingdom Come: Deliverance.
Edit: I also just remembered the hacking system in BioShock had a very mini-game feel to it. I had a lot of fun with those too.

I believe the correct answer is the loading screen minigames from Dragon Ball Budokai Tenkaichi 1-3

Many people love Stardew Valley so I gave it a go. It was fun for a couple of hours, but it doesn't really have any depth. But you can go to the tavern and hop on an arcade machine with a really fun minimalist twin-stick shooter called Journey of the Prairie King that's actually pretty great. It's fast paced and unforgiving, and I spent more time playing that than on actual farming and what not.

I'd throw Crash Bandicoot showing up in Uncharted 4 into the mix, with an honorable mention for Nathan's Nerf gun setup in his attic.

Maybe neither of those count, as they're not complete in and of themselves? But I think they count in the spirit of the question asked.

Not sure if this totally counts but my favorite is the Chao raising systems in the Sonic Adventure games

I really enjoyed the classic wolfenstein levels inside the new wolfenstein games

For minigames as "games within the game" (e.g., GTA has a lot of these like pool, golf, etc.,) throw another one up for Witcher 3's Gwent!

For minigames as representations of some other mechanic (e.g., hacking, lock picking,) I remember liking the hacking in Deus Ex: Mankind Divided. Reminded me of hacking in EVE: Online.

Probe scanning was awesome in EVE too—at least...it was a decade or so ago. Who knows if it's still the same now doe? Not me.

I like how the Yakuza/Like a Dragon is jam packed with mini-games. Sega even puts classic arcade games in it. But I feel like bang for your buck, you're going to have the most mini games in the Yakuza game than you will in any other game.

I don't know if it counts but Red Dead Redemption introduced me to Liar Dice! Now I freakin' love Liar Dice!

I can't think of any memorable "hacking" type ones as they all just become a chore by the end. Fable II has some wood chopping, pie making and lute playing that wasn't so bad if you can get a high multiplier going.

As for actual "games within a game" then Shenmue series has many gambling and arcade machines. Roll it On Top, Lucky Hit, Darts and then Arterburner, Space Barrier, Outrun and Hang On.

Doesn't completely fit your description of minigames, but I spent days playing Casino card games with Luigi in Super Mario 64 DS. And the Hide and Seek game was great too.

Those minigames are separate from the main game though.

FICSIT Productivity Packer Deluxe (Satisfactory). It's a fun little game available at The HUB once it is fully upgraded. You take Tetris-like pieces and arrange them to fit within a square, completing as many squares as you can within the time limit to determine your score.

Then again, the real mini game in Satisfactory is the planning that goes into your factory while you are not playing the game. It's the game that just keeps on giving.

Anaconda in TimeSplitters 2, mostly for the incredibly catchy music.

Splatoon 2's rhythm game. I am forever angy that splat 3 has no rhythm mini game

Galaxian as Ridge Racer loaded on the PlayStation.

The FF9 tetra master card games and chocobo digging

“The coin game” has some of the best designed minigames ive ever seen!

The game is essentially just a collection of minigames, but it is put together incredibly well

The arcade games in Grand Theft Auto: San Andreas

Triple Triad from Final Fantasy VIII

Chocobo Hot and Cold, & Tetra Master from Final Fantasy IX