I havent played much handheld stuff prior to Switch, but one i really enjoyed was A Link Between Worlds on 3ds!!!

Game Boy Tetris is So Fun
Mario Kart DS, New Super Mario Bros (DS) were also fun
Nintendogs was very fun.
Smash for 3DS had better Single Player Content that Smash for Wii U, so yeah that was a fun game too.
Fun is the only word i can use to describe them, but if i were to say which one is best i would say Tetris due to it's cultural impact. Quote, on the impact of Tetris for Game Boy: "Tetris made the Game Boy, and the Game Boy made Tetris."

Gameboy Advance SP for me, with Legacy of Goku/Pokemon Silver

My favorite is Fire Emblem Awakening for the 3DS. It's the only handheld game I consider one of my favorites of all time. It's a game where the story and gameplay mechanics work in tandem to make each other better. Nothing else comes close.

Does it have to be handheld exclusive?

My favorite handheld game I've ever played, was Chrono Trigger on the DS..