I am running a Joomla 3.3.6 site in a hosted environment (Startlogic) and I see that my File Cache Handler option is no longer available. It used to be an option at some point and now it is gone. Cache_Lite is currently my only option. I have found that for my site the File Cache Handler was much faster so it would be nice to have it back.
I have a cache directory and it is set to 755 permissions.
This article was helpful but not really my situation : Joomla cache handler only shows memcached