Dragon Warrior Monsters

From Wikipedia:

Dragon Warrior Monsters (known as Dragon Quest Monsters (ドラゴンクエストモンスターズ Doragon Kuesuto Monsutāzu) in Japan) is the first video game in the Dragon Quest Monsters series. It was released in Japan by Enix on September 25, 1998 and co-published by Eidos Interactive in North America on December 31, 1999. It was released for the Game Boy Color before the console itself was released; however, the cartridge is backward compatible with the older Game Boy console in black-and-white color. The game was remade for the PlayStation as Dragon Quest Monsters 1+2.

This game is the first Dragon Warrior Monsters game to be released for the Game Boy Color. It features the characters Terry and Milayou from Dragon Quest VI when they were children, before the events in Dragon Quest VI. Dragon Warrior Monsters is often compared to the Pokémon series due to their similar gameplay. Critics set the games apart by noting the in-depth breeding system seen in this game.

The game follows Terry as he attempts to save his sister. Instead of fighting in battle, Terry is able to recruit monsters into his party. When fighting random battles, Terry can set out meat, which may coax the monster into joining the party at the end of the battle. Terry can also breed two monsters, combining them into a new, stronger monster. Excess monsters can be placed on the Farm, a storage place for monsters. They can be picked up or dropped off and there is an option to put monsters at the farm to sleep, which allows them to maintain their wilderness level, but not grow in level. There can be 19 awake and 19 asleep monsters at any one time. A monster's wilderness level lowers as Terry trains them, which makes them more likely to participate in battle. By leaving monsters at the farm awake, they become more wild.