I have a fix.
Problem.
After downloading (3 hours) and installing El Cap on MacBook Air 13 inch early 2015 there are two requests for login. The first request has a fuzzy image of Yosemite valley in the background. Then you are requested to login again. This second login is on a grey screen. The progress bar stops without completion.
No change for 8 hours.
When performing the Recovery mode restart below (R Restart) Do not bother with the option to install El Cap again. You already have a good installation. We just need to clear some files that are stopping the installation from loading.
Solution
Shut down.
Restart holding down the R key. Release R key when you see the apple logo.
Login when requested.
Desktop should load. If it does not, shut down and repeat.
When desktop loads, GO/LIBRARY and drag folders; CACHES, STARTUP ASSISTANTS, STARTUP DAEMONS to trash
(Hold down the option key to reveal the LIBRARY)
MACINTOSH HD/LIBRARY and drag folders; CACHES, STARTUP ASSISTANTS, STARTUP DAEMON and COOKIES to trash.
Shutdown.
Restart holding down CMD OPTION P R. Continue holding down the CMD OPTION P R keys until you hear a second 'bong'. Release keys
Login when requested.
Your desktop should now load.
Empty trash.