What permissions does root give to an app?

From my understanding and past experience programming Android apps in rooted phones, getting root permissions in an app is to have access to everything.

Once a phone is rooted, you, the user, have access to the entire file system¹, which allows anything to happen with it, including apps modifying their own permissions².

Android is built on Linux, which means that su, or the root user, is the master user. Root user has the ability to do anything they want³. Apps granted root permission can add, modify, or delete things as they please throughout the file system. Along with that, they have access to the APIs that control the hardware of the phone, which means that they are capable of doing anything without your knowing, among other things (reason 3).

TL;DR Root has access to everything and can do anything.