A few favorites I haven't seen anybody else mention yet --
- Chulip, the kissing RPG about kissing where you have to try to kiss everybody in town... even the sewer residents. It's on PS2.
- Moon Remix RPG Adventure, on Switch, PC, and PS4. Calls itself "the anti-RPG". You're the guy who has to fix everything after some hero stormed through the world on some quest and wrecked everything in the process.
- Freshly-Picked Tingle's Rosy Rupeeland, on DS. One of the very weirdest, funniest, weirdest, most visually unique, and definitely weirdest RPGs of all time. And it's a Zelda spinoff! Tingle the man-elf is on an adventure all his own, and though it does feature some "combat", trust me: if you're looking for a weird RPG, this is DEFINITELY what you're looking for.
Two quick honorable mentions, both for Wii:
Opoona is a really cute RPG about a blobby little alien guy who crashes on a planet and has to get a job to find his blob sister. Weird and a lot of fun, but definitely has turn-based combat so not quite what you've asked for.
Little King Story is an endlessly charming RPG/RTS/Life Sim/Pikmin-like that doesn't get anywhere near the love and discussion it deserves. A kid finds a magic crown that makes everyone magically bow to his will, so what else is there to do but conquer the nearby monster-filled kingdoms one by one until you rule all!? Visually and especially musically, one of the prettiest games on the system, and just an insane amount of stuff to do and see. Not really an RPG and definitely has combat, but if there's one game out of all of these I wish more people would play, it's Little King Story. (The Wii version if at all possible -- the PSL remake-quel was an admirable attempt to keep the series rolling, but man oh man, they butchered the aesthetic and slathered the whole experience in microtransactions. Then it still sold worse than the original! Shame... A modern version could be incredible.)