The last two updates of WordPress (4+) have produced a redirect loop when attempting to log into the wp-admin page. I went through numerous solutions found online and the one that worked for me was to change the permissions on the wp-login.php file (in the root directory) to 644. It seems that when doing the update, WordPress changes the permission on this file. Once I changed it to 644 in my FTP client (filezilla), it instantly worked! So, this latest update (4.4.2) created the same issue and I had it back up and running in 10 seconds!
UPDATE: Upgrading to WP 4.5 produced an error on the front end and it ended up being the same issue. The admin section lost all of its CSS and I couldn’t get to any page I tried to click on.
Change all the permissions of the FILES under the wp-admin folder to 644. WP changed them to 777. Change your index.php file under the root directory to 644 as well.