Q: Macbook Air 2013 with El Capitan was 10.11.5 went to 10.11.6  - loops at login

Hello buzzstarlog,

Thanks for that info and choosing the Apple Support Communities. I know how important getting your MacBook Air working properly again is to you! If I understand correctly, you are able to start up to the login screen, but not able to finish the login procedure, and even some troubleshooting has not helped. At this point I suggest these steps to resolve your issue:

1. Use Safe Mode to start up your Mac so that it performs certain checks, delete system cache files, verifies your startup disk, and attempts to repair directory issues if needed.

Try safe mode if your Mac doesn't finish starting up
Limited graphics performance in OS X recovery or safe mode

2. If Safe Mode didn’t help, you maybe able to use Disk Utility in OS X Recovery Mode to make a disk image of your internal hard drive to backup your personal data:

OS X El Capitan: Alternatives for backing up your Mac
Disk Utility (El Capitan): Create a disk image

3. If you're able to backup your personal data, you may need to try erasing the internal hard drive and reinstall OS X using these steps:

How to reinstall OS X

Cheers!