Rakuen.

Rakuen is a story-based adventure, but there are no battles. It functions like a mix between Maniac Mansion, To the Moon, and The Legend of Zelda: Link to the Past (in terms of layout, not in terms of fighting). The hospital itself has a lot of little puzzles and mysteries that evolve throughout the course of the game, and you can gradually help improve the quality of the place as you progress. And of course, a big part of the game is uncovering the stories behind the patients in the hospital, and the secrets of the world around you.