Crash Bandicoot Blast
This mini-game was developed by Vicarious Visions, and it's accessible by linking a Game Boy Advance to a GameCube while it's running Crash Bandicoot: The Wrath of Cortex in it.
The goal is to aim Crash's fruit bazooka to win points by shooting jumping pictures of Cortex, N. Gin, Tiny, Dingodile, and Uka Uka. Do not shoot Aku Aku, Coco or Polar, or you'll lose points. You can shoot crates to gain more ammo, preventing you from getting a Game Over. Once you get enough points, you'll proceed to the next round. The speed of the targets increases as the rounds pass by.
Crash Party / Crash Party USA
This secret mini-game can be accessed in the Game Boy Advance version of Crash Nitro Kart and in Spyro Orange (a.k.a. Spyro Fusion). In the latter, "USA" was added to the title for some reason, but the game is exactly the same otherwise. To access it, just hold down the L and R buttons while turning the console on.
The goal of this game is to make Crash boogie with rhythm. To do so, press the buttons that appear on the screen whenever they're behind the green circle. Sometimes you have to switch tracks with the D-Pad. A similar mini-game called Spyro Party USA appears in Crash Bandicoot Orange (a.k.a. Crash Fusion), which is basically the same thing, with Spyro instead of Crash.
In Crash Nitro Kart, if you manage to beat this game's 10 rounds, you get 3 immediate missiles for a race against Emperor Velo.
PocketStation Virtual Pet
This virtual pet can be accessed through the PocketStation, a peripheral for the original PlayStation released only in Japan. It's basically a small device that acts like a Memory Card while sporting an LCD screen and buttons. It's capable of storing interactive applications, and the Crash Bandicoot virtual pet is one of them.
To access this app, connect a Pocketstation to a NTSC console while running a Japanese copy of Crash Bandicoot 3. The main goal of the app is to collect photos of Crash. To do this, you can play various mini-games, such as the virtual pet, dancing sessions, boulder chases, and a type of jigsaw puzzle.
(Thanks to FakeNina for the screenshots.)