This really, really interests me as well, and I've written my iOS apps the same way too. They have no server side, no real dependencies, all client based and fully automated internally - so as well as being fully contained they schedule required notifications etc into the future without my intervention. As long as the user runs the app once about every four months (which isn't an issue given they get many notifications during that four months) then the app will work for a very long time and bring in passive revenue. I consider this a holy grail of software.
I like pure solar powered calculators for the same reason. If you were to send one back in time it would continue to work and be useful without requiring intervention or dying.
Yes, they certainly can. Apple allows any apps notifications to be switched off at the OS level - it isn't optional. My users love the notifications though, and they commonly customise to get _more_ notifications. The reason it is 4 months is because iOS only allows 64 notifications to be scheduled in advance for any app, and if they want users can configure to schedule that many in a 4 month period.
I like pure solar powered calculators for the same reason. If you were to send one back in time it would continue to work and be useful without requiring intervention or dying.