I started having this problem last night after plugging my phone into the charger. The battery was low (35%) so once it was charging, the phone decided to update some apps. From that moment forward, my phone would do the black screen with spinning wheel about every 3 minutes - almost too fast for me to even get a successful backup. The phone would also get quite hot. So I suspected that it was related to an app update.
Restore in iTunes worked for me - the phone stopped misbehaving. BUT soon after restoring from backup, as soon as all the apps had finished downloading & installing again, the problem returned (restoring from backup does not restore the apps - just app placeholders and app settings - the apps then need to download/install again from the App Store). So from that point I knew it was an app causing the issue. This makes sense, because the problem started after a bunch of apps updated the night before. So I deleted about 20 apps (non-Apple ones like Shazam, Spotify, various airline/airpoints apps, about half of what I could see - then I stopped to see if the problem had gone. And yes, the problem had gone away. So I did a Restore from backup again, with the intention of reproducing the problem (and then uninstalling apps one by one, waiting a few minutes between each one to see if the problem had gone away). But this time I had a better idea and when the apps were reinstalling, I quickly tapped on any apps that were downloading/installing, to pause them, and the problem has not returned. This means one of two things:
A) the problem is either Heatmiser (app to control my central heating), Shazam, Headspace, or Trello (the only ones I managed to catch before they installed)
B) In the 18 hours or so between the time when I put my phone on charge at night, and when the problem went away, an app developer released a bug-fix for the problem they created 18 hours earlier, and after restoring my phone, perhaps it downloaded the latest version which fixed the issue. E.g. whatever app was at fault, got updated during the time I was messing around with my restores (but it must've happened within a 15 minute window, so quite a big coincidence).
My point is, this can be an iOS issue, but it can also be an app issue. My symptom was regular black screen with spinning wheel followed by asking me for passcode or Touch ID (this means it's not a full reboot - the iPhone will not accept touch ID after a reboot) - the phone would get hot - and during each reset cycle the phone would make that chime it makes when you plug in the charger. Any text or email you were writing, would be lost during the reset. Full power offs and stuff did not help at all of course.
So, if you have this issue, before you erase/restore everything, you might want to first try:
1) do a full backup in iTunes - make sure you tick "Encrypted backups" as this will mean that iTunes stores all your passwords and some other extra stuff in the backup file. It means when you restore, you'll have less work to do, to get your phone back to its original state. BTW, I noticed that even if the phone did its reset thing in the middle of an iTunes backup, iTunes seemed to be able to complete the backup after the reset finished. But out of paranoia I kept trying until I managed to time it right and get a full backup between resets.
2) delete about half of your 3rd party apps from the phone (or if you like, delete them one at a time and just wait between resets). If you can't be bothered staring at your phone to see if it's still resetting, just open up Messages and type in an SMS to someone but don't hit send. Then when you come back to your phone you can tell if it's reset, because it will be at the home screen and the unsent SMS text will be lost. Once you find the bad app, maybe contact the app developer and tell them what's happening, or try reinstalling it to see if you can reliably make the problem reappear and disappear by installing then reinstalling the app. If you get to zero 3rd party apps and the problem hasn't gone away, THEN...
3) so you couldn't fix the problem by finding a flawed app... so maybe it is an iOS issue. In that case, Erase/Restore via iTunes (assuming you backed up in step 1!!) then Restore from Backup. If the problem is still present then in my mind it has to be an actual iOS bug (which some of the first original posters in this thread experienced with iOS10), or an app bug that hasn't been fixed yet.
I will edit/post again when I later have some more time to unpause the install of those remaining 3 apps that are paused, but right now I don't have time to risk it as I need my phone for work for the next few hours.