Neko Atsume is an iPhone app whose whole purpose is to attract cats to your virtual yard. I still don't fully know how to play the game and there isn't a lick of English in it. Despite this, I'm addicted.
I've had this app on my phone for two months and find I check it nearly every day, leaving out toys and feeding virtual cats. It harkens back to Tamagotchi except with even less stakes, and the music is soothing in a bit crush kind of way.
Here's where to find it: