Generally speaking, I like Fighting Games, RPGs, Visual Novels, and Action/Adventure games. Tons of bonus points are issued for good stories, interesting mechanics, and a neat aesthetic.
Due to this, I generally prefer Japanese games, but the West does seem to be upping its quality in recent years with things like the Arkham games, Injustice and a bunch of neat indie titles to go along with some of the better classics from this side of the international dateline like the Ultima series. And I am also big on Point and Click Adventures, and I can't think of anything quite like King's Quest out Grim Fandango from Japan, off the top of my head (though I'd be interested to hear of them).
Edit:Mmaaaybe Dangan Ronpa or Ace Attorney are kinda sorta that kimd pf game, if you squint.