Vib Ribbon on Playstation. JP only I think but it's pretty well known
Some other games only released in Japan:
Takeshi no Chousenjou (Famicom)
Famous comedian and film director Takeshi Kitano doing his own game, and trying to break every rules about video games along the way.
Utsurun Desu - Kawauso Hawaii e iku (Famicom)
It's your typical plateformer, but full of "breaking the fourth wall" type of jokes, like false title screen, gaining suddenly the ability to walk on the background, and more...
Hayauchi Sokutei Game (Famicom)
Best game you'll ever play.
Anybody can read French? because http://oujevipo.fr/ is the best place to find hundreds of weird video games. I recommend the 2013 and 2012 end of the year top lists.
A few years ago I made a RPGMaker game with a bunch of other people (like Mortis Ghost, who a game called Off that went pretty viral on Tumblr and that is worth mentioning here : http://forum.starmen.net/forum/Fan/Games/OFF-by-Mortis-Ghost/page/1/) called "DarkSoule.Ace2", I guess it could be described as some sort of demented and very ugly Wario Ware game going in numerous random direction every few seconds, where you only very slowly realize the existence of a plotline and that is endless and never the same every time you play it. It was shown on french TV at some point :