OK, what you have is unfortunately not related to Akeeba Backup. It's either a problem with your user session or a problem with Joomla!'s access control setup. Neither of these has to do with our code. I can try to help you, but please understand that I'm trying to help you with a Joomla! issue so your mileage may vary!
First try removing all cookies from your browser that have to do with your site. If you are unsure you can simply remove all cookies from your browser. Then retry logging in to your back-end.
If you still don't see your back-end menu, it's possible that you accidentally unpublished the system menu module. Go to Extensions, Module Manager (the /administrator/index.php?option=com_modules URL on your site) and click on the first Filter on the left hand side of the page which currently reads "Site". Change it to Administrator. Find the "Admin Menu", "Admin Submenu", "Title" and "Toolbar" modules and make sure they are published and their access is set to Registered. Then make sure that the position of each of these modules is:
"Admin Menu" --> menu
"Admin Submenu" --> submenu
"Title" --> title
"Toolbar" --> toolbar
Nicholas K. Dionysopoulos
Lead Developer and Director
🇬🇷Greek: native 🇬🇧English: excellent 🇫🇷French: basic • 🕐 My time zone is Europe / Athens
Please keep in mind my timezone and cultural differences when reading my replies. Thank you!