Diablo 3 comes to mind as the canonical example of this, but it's a PC/console game rather than mobile. Single player mode is still run by a remote server, and if you have a bad internet connection or they have a server outage (which has happened many times) it will be unplayable.
Blizzard has been much better than most companies about keeping its online infrastructure up and running long after a game loses popularity, but even they won't be around forever.
On the mobile side I don't think that's a common strategy (Candy Crush and Angry Birds aren't going to stop working), but the purely multiplayer games like Ingress and Clash of Clans will certainly shut down someday.
I was asking for examples of mobile apps. There are lots of reasons that the "thin client even for singleplayer" strategy is less feasible for mobile games than for PC games.
But still, I didn't buy Diablo 3, and this was one of the reasons why.
Some examples that come to mind are Candy Crush, Marvel Puzzle Quest, Farmville, and Angry Birds. But these aren't mobile-exclusive, and they phone home to retrieve your saved game state from a database.
I'm not aware of an Android-exclusive or iOS-exclusive game that is immersive and not casual. Certainly nothing on even a Diablo-level. Anything close is multi-platform effort.
I just downloaded Angry Birds and it works fine in Airplane Mode. If Rovio goes out of business and it fails to sync my saved game to a new phone, I won't be that sad. I wouldn't say it falls under the "thin client making API calls" style software that cableshaft mentioned.
Blizzard has been much better than most companies about keeping its online infrastructure up and running long after a game loses popularity, but even they won't be around forever.
On the mobile side I don't think that's a common strategy (Candy Crush and Angry Birds aren't going to stop working), but the purely multiplayer games like Ingress and Clash of Clans will certainly shut down someday.