View list of registered document providers
I assume were referring to the file types the different apps can open.

On a rooted device you can get the list the following way:

The mime types and the applications for the default user are stored in the file /data/system/users/0/package-restrictions.xml

Therefore you can execute the following PC to get the full list on a smartphone connected via USB (an enabled ADB):

adb shell "su -c 'cat /data/system/users/0/package-restrictions.xml'"