A couple of games that have come into my mind:
Rhem - mostly because it was just too hard for me to play without a walkthrough.
Uru - because it had jumping in it that I was unable to do but I may try the game again someday.
An older game I wouldn't call classic but which seemed to be very well liked, was just mentioned in a different thread - Scratches. I played the game twice. I guess it didn't make much of an impression because I just had to Google the ending and still don't really remember it. There are some small parts of the game I do remember but, for me, overrated!