I don't get it. 0777 is already the broadest permission you can give. So what is this minimum required permission?
I haven't done much FTP or shared hosts since, maybe, 7-8 years, but I remember Wordpress has the same 0777 is bad practice for many years and hasn't changed.
777 is never a good option.
[1]: http://codex.wordpress.org/Changing_File_Permissions#Shared_...
Anyway, these days, for a couple dollars you can have a really really good cloud hosting service.