Well, there's the badges for some icons, and there are the widgets that run in the screen to the left of the Home Screen (whatever that's called) that do refresh - and one might assume that the new iOS Weather widget for the Home Screen does (though no guidance as to how often), and their Photo one cycles thru your photo library, but they certainly don't seem to be encouraging 3rd Party ones to have active connections. This, I appreciate, may be a battery issue (to which Apple is deservedly sensitive) or has privacy concerns, but that should be up to the user to decide as to whether they are willing on taking hits on those.
I mean, if Apple was truly concerned over battery or privacy issues, they would have banished the Facebook app years ago... 😉