Your upgrade to Joomla! 3.3.6 failed. Since the libraries/fof directory already existed from a very old version of Akeeba Backup (3.6 I guess?) the Joomla! upgrade didn't replace it. Since the very old version of FOF is definitely not compatible with Joomla! 3 you got a problem.
Do not worry. You can fix it easily.
1. Download the installation package of Joomla! 3.3.6 from http://joomla.org/download.html
2. Extract the ZIP package locally
3. Delete the libraries/fof (f-oh-f) BUT NOT THE f0f (f-zero-f) -- THIS IS EXTREMELY IMPORTANT -- directory from your LIVE site
4. Upload the libraries/fof directory you extracted in step 2 to your LIVE site's libraries/fof (f-oh-f) BUT NOT REPLACING OR PUTTING INSIDE THE f0f (f-zero-f) -- THIS IS EXTREMELY IMPORTANT -- directory
Please keep this in mind regarding the directory names in your site's libraries directory:
fof (f-oh-f). This contains the FOF library. More than a year ago all FOF based software would install the FOF library over there. Since Joomla! 3.2.1 FOF is part of Joomla! itself and no FOF-based software will replace this directory as long as they follow my very clear instructions on installing the FOF library. The class prefix of the library in this directory is FOF (F-oh-F). Typically, the FOF library version in this directory is 1 to 6 months old due to how Joomla! release management works.
f0f (f-zero-f). This directory contains the bleeding edge of the FOF library. It is installed by FOF-based components which want to use the latest and greatest features without waiting for a new Joomla! release. The class prefix is F0F (F-Zero-F). It is NOT used by Joomla! itself and we encourage other developers to use this version when they need access to bug fixes and new features added in newer FOF versions.
As a result we only touch the libraries/f0f (zero) directory with our software, NOT the fof (oh) directory which is part of Joomla! itself. If a developer replaces the libraries/fof (oh) directory in Joomla! 3.2.1 or later we let them know that what they're doing is wrong.
I hope this information helps.
Nicholas K. Dionysopoulos
Lead Developer and Director
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