The laptop, which is relatively new, has 2 outputs (HDMI and analog) plus the built-in screen. Has anyone done this before? Will AMD / ATI's new drivers that support the 5xxx series maybe bring support for more than 2 screens to older cards? Is there something I need to do that will enable this?
You can use a small external device called Matrox Dualhead2Go or Matrox Triplehead2Go for this. It plugs into the HDMI output and splits the image across two or three displays. You should be able to use one image from the display on your laptop screen and split the image from the external output on two or three screens. Depending on which device you use, you can populate three or four screens in total.
The limitation is the image displayed on the Triplehead2Go or Dualhead2Go acts as a single image for Windows. This may or may not be a problem depending on what you want to achieve. You can place your windows and applications as needed, but any fullscreen elements will span across both / all thee external displayes.
While this may be not what you are about, without an external device you are out of luck. The HD 3600 features only Two independent display controllers, not three, same as most display adapters.