Gonna have to go with Sega Saturn de Hakken! Tamagotchi Park on this one. Basically, the gameplay matches that of the first Game Boy game, except with some notable differences.
I quickly wanted to bring up the character lineup before discussing why this one is my favorite. First, the G1 and G2 lineups are no longer combined into one. This alone is a VERY welcome change - the G1 lineup is what you get in the initially available egg, while the G2 unlocks later on, as do two more. The first one is the Angel which includes the same characters as the keychain (Even Lucky Unchikun), and the second is a Sega themed egg... which is one of the two big reasons why this is my favorite of the video games.
The Sega egg basically hatches into a small computer ROM chip (ChibiROMtchi) before evolving into a bigger one (ROMtchi). After this, it will evolve into one of two teens as usual... those being Disctchi (Regular CD, because the Saturn CDs were no different from the others in appearance, let's face it) or Cartritchi (Genesis / Mega Drive cartridge). Both Disctchi and Cartritchi have some VERY cute animations... but it's not until they grow into one of the four adult forms that things get AWESOME.
Long story short, the adults and the secret character on this egg are all based on actual Sega characters. Alextchi (Alex Kidd), Opaopatchi (Opa-Opa from Fantasy Zone), Pengotchi (Pengo), and Flickytchi (One of the Flickies from Sonic 3D Blast), with Sonictchi (No need to say who that one is based on!) being the "not so secret" character - I say this because Sonictchi and Opaopatchi are prominently featured in the advertising materials for the game. In fact, they're both featured alongside the main cast of the G1, G2, and Angel in the game's TV commercial.
Aside from the awesomeness that is the Sega egg, the other reason I love this game so much is because it has the best graphics out of the entire vintage game lineup - not even Hoshi de Hakken! Tamagotchi (For the PlayStation) and Minna de Tamagotchi World 64 (For the Nintendo 64, which is a Mario Party type game with light virtual pet elements - nowhere near the same as the Game Boy games, this one, and Hoshi de Hakken are in gameplay) come close visually. All of the characters from all four eggs are extremely well animated - even basic animations such as eating (Which in the Game Boy games and Hoshi de Hakken only have the usual two frames) are given attention to detail. And of course, the Sega egg's adults carry the same mannerisms of the characters that they're based on... gotta love Opaopatchi growing his feet out on occasion!
So yeah... Tamagotchi Park is my favorite of the Tamagotchi video games. No contest.
(Also, I recently did a YouTube video of the game's character viewer mode, in which I went through the animations of all nine Sega egg characters as well as Lucky Unchikun. If the mods are okay with it, I'll post the link... but until then, I'm going to play it safe.)