Quite the old game here, but still quite awesome. I haven't played it, but I've seen multiple playthroughs...
All slightly different.
Yume Nikki, Dream Diary in Japanese, is a creepy horror-ish exploration RPG. There is no main plot, but a gigantic fanfiction. The main goal of the game is to collect all of the possible power-ups possible in the game, and then find out what you need to do with them. You are not given where or how to find these power-ups, or that you're even supposed to find them; you're just thrown in the game, in a little girl's appartment...
You play as a character named Madotsuki, a shy girl who never wants to go outside. The reason? No one knows. All she wants to do is go to sleep, and as she does, you explore her dreams.
Disclaimer
This game is known for its creepy and disturbing moments. If you do not enjoy creepy and disturbing moments, do not play this game.
Edit: Also, stamps;
Edit Edit: Actually, considering Yume Nikki is all about first experiences with things... I'll only show you this Let's Play, in case you won't play it.
LoudMan01's Let's Play of Yume Nikki;