I'm developing for Android > 5 so I don't need to use the appcompat library. Unfortunately, all the examples on this forum to use a toolbar with a menu and a button use ACTOOLBAR, which needs appcompat. Is there an example of a toolbar which follows the material design and does not need the appcompat library?
I don't think so, otherwise I wouldn't suggested it.
AppCompat doesn't change the views on Android 5+. The compatibility layer is mainly for older devices.