Hi David,
If you’re talking about the Maximize command as found in the Customize Palettes/Commands window, I can concur that it doesn’t work as you desire in either R19 or R20, either by adding the icon to a layout, or a keyboard shortcut for it. In fact, I can’t see that it does anything. However, as the Help describes (search for “maximize”), with the cursor over the desired window, Ctrl/Cmd + Tab will accomplish what I think you want. The Customize Palettes window lists this behavior under “Full-Screen Mode” (and, interestingly enough, only lists Cmd+Tab on my Mac, but works as Ctrl+Tab). If you’re on a PC, I assume that it should work as well.
Hope this helps.
Edit: I forgot to add that the Full-Screen Mode shortcut can be changed to whatever you want (as long as its available).