This may be a bit of a conventional answer, but it's got to be Yume Nikki. That game mesmerized me the first time that I played it and after all this time, even after played many of the games that it inspired, I still find I love the game. In fact, when I made a top 10 list of games a while back, I put it at #5.

I usually see the game listed as a "horror" game but I honestly don't quite agree with that assessment. Yes, there are some unsettling aspects to the game, but I see it more like an amusement park attraction, where each area is a set piece you walk around and observe that allows you to see part of what lurks in one person's mind. To be honest, I actually find much of the game quite calming. My favorite area is the Witch's Island, named for the effect you find there.