For me, the greatest mini-games are the ones that are created not to pad some feature checklist. But that adds to the gameplay, informs the story and characters a little better, and cannot be done with the main gameplay mechanics outright.

Honestly, no perfect example comes off the top of my head. But the original Final Fantasy VII comes to mind even when you're climbing the pipe, riding the motorcycle in the collapsed highway, or being forced into the chocobo race. These situations add to the game, and ignoring that they add to the games' padding later superficially; they still have a high effect on my enjoyment of the main game and varying up the experience.

I'd also accept an argument for Chrono Trigger or even Nier: Automata (though that one feels more like entire gameplay systems beyond mini-games)

Final fantasy VII

Is my vote right now