Automatic downloading of updates
This issue is fairly close to #16 (closed) and #106 (closed), but based on what F-Droid can currently do, and most likely much easier to implement.
Currently, F-Droid checks for updates, and then notifies about which apps have been updated.
I envision a setting that would make F-Droid start downloads for updates in the background right after an update check, and then showing the "Download complete, tap to update" notifications for the updated app(s), allowing the user to start the update of that specific app with a single tap.
This may require simple caching to prevent unnecessary redownloading in case of high update check frequencies (such as hourly) with the user not responding quickly enough (say, when asleep).