Remove the negative cable from the battery and hook up the positive cable. Now, remove the fuses from the fuse box. Close the doors and touch the negative cable to the battery negative post. If you hear the clicking, it has to be something to do with the starter.
The starter gets power directly from the battery. It's possible (I suppose) that the wire to the solenoid has somehow shorted and gets power as soon as the battery cable is installed.
If nothing happens when you touch the the cable end to the negative terminal, install the fuses - one at a time - until the clicking can be heard again. Run the electrics back from that fuse until you discover the source of your issue.
Scott Stuchell