I work at a relatively high application scaling, due to which I cannot see the current vertex/face/tri counts in the application header. At the same time I find myself never quite needing the Layout or Scene selectors. Is there a way to remove these, so I can have what I need and lose what I do not?
Press ⎇ AltF10, this is the shortcut for Toggle Fullscreen Area, which can be found in the View menu. As you loose the menus in this mode you either need to use ⎇ AltF10 or ⎈ Ctrl - ← Left arrow or → Right arrow to get everything back. This differs from maximise area as it also removes the info editor and headers.
If you also use ⎇ AltF11 to go into fullscreen mode you also loose the window border and get nothing but the 3dview on your monitor.
An older answer provides an addon that displays mesh component counts. A small adjustment will enable it to display in the properties region so that you can see it while in full screen mode, using N to show/hide it.
class Properties_meshinfo(bpy.types.Panel): and change the following lines to place it in the 3d view.
class Properties_meshinfo(bpy.types.Panel): bl_label = "Mesh Information" bl_space_type = "VIEW_3D" bl_region_type = "UI"