How can I get safari to show me the pixel dimensions of the browser window, without an extension?
You have to have the element inspector open.  So with element inspector open, then, In Safari, when you hover over an element, it shows the pixels. 

And you can get it to show the pixels while you resize a window, to do that you click the crosshair icon in the inspector, then hover over the background so it shows the pixels, then move the mouse across so it still shows the pixels, and resize the window and it will update the pixel dimensions.  

Note that in Chrome, if you have the inspector open, and resize, then it shows you the pixel dimensions while you resize.