From StrategyWiki, the free strategy guide and walkthrough wiki

Professor Samuel Oak - The leading authority of Pokémon research. He serves as the tutor for teaching you basic gameplay and techniques at his lab.

Gary Oak with evolved Pokémon - As the final stage before battling Mewtwo, Gary battles you a second time.

Cassidy and Butch - Rivals of Jessie and James, respectively, who work for Team Rocket. The two of them battle you in the Spa Service as a bonus challenge.

Nurse Joy - A Pokémon nurse who has identical-looking female relatives (all are nurses with same name). She is seen at the Pokémon Center to modify the game settings.

Officer Jenny - A policewoman who has identical-looking female relatives (all are all policewomen with same name). She is seen at the Pokémon Center to modify the game settings.

Delia Ketchum - Ash's loving mother. She can be seen and heard in cutscenes after clearing the 1P Stadium at any difficulty level.

Jigglypuff - A Pokémon with a desire to sing. It is seen or heard at all the modes except 1P Stadium and 2P Stadium.

Togepi - Misty's Pokémon. It can be seen and heard when playing Super Easy Mode in the Mimic Mansion, Time Zone, Marathon, and in cutscenes after clearing the 1P Stadium at any difficulty level.