Remove the "Add" functionality in Django admin
Satya's suggestion of setting max_num=0 works perfectly. 
Per the Django docs on the ModelForm class:

For users with JavaScript-enabled browsers, an "Add another" link is provided to enable any number of additional inlines to be added in addition to those provided as a result of the extra argument.
The dynamic link will not appear if the number of currently displayed forms exceeds max_num, or if the user does not have JavaScript enabled.

and

As with regular formsets, you can use the max_num and extra parameters to  modelformset_factory to limit the number of extra forms displayed.
max_num does not prevent existing objects from being displayed
Is there a way to remove the "Add" functionality on the Django admin site? For certain entities, I only want the Django admin to be able to view them or change existing ones, but not add new ones.