2. I'm working on an app that constantly records audio in the background and later allows me to search for things I might have said throughout the day
3. Apps that track location long-term, like the defunct Moves app. Sure it kills the battery and is a major privacy concern so it should be opt-in, but I personally love this functionality and I'm sad there doesn't seem to be a modern equivalent (Google Maps location history is less accurate and there's no good API). Apps that record location short-term like MapMyRun do not solve the same problem.
... I'm sure there's other use cases. I kind of feel like the only developers allowed to really innovate on mobile anymore are Google (and Apple) themselves. Bad developers abusing privacy/battery/data have made it so Google/Apple have to lock down anything interesting. It's frustrating, because the current state of affairs is definitely better than the free-for-all it used to be, but I wish I had more control to override what my own device is able to do.