I recently got a refurb Elitedesk 800 G3 for the low price of $60. (The intent was to use it as a host machine for my SQRL Acorn.) It was easy to tell what made those machines so cheap, the GPU built into the CPU is basically garbage (except for its Quick Sync functionality) and the 180W proprietary PSU is too small to run anything but the most basic of GPUs.
The PSU has 2x 4 pin connectors for 12V. The 12V, unlike an ATX PSU, is always on. Then there's a 7 pin connector for control signals.
1, black - ground.
2, violet - POWER METER, gives an analog voltage in proportion to the output current.
3, gray - PSU-ID, 383k to 12V for 180W. This likely is different for different wattages.
4, green - PS-ON. Unlike an ATX PSU, it just tells the PSU to go into full power mode when grounded. The 12V output doesn't change in voltage at all.
5, black - ground.
6, white - FAN TACH, just a passthrough to the internal fan.
7, red - FAN CMD, goes to some 8 pin chip that takes care of fan control.
While I haven't tested it, I assume a resistor to pin 3 and something to provide a fan signal on pin 6 are all that's necessary to replace the PSU with something bigger. If the intent is to install a GPU, a switching circuit will be needed for the GPU 12V to ensure correct power sequencing.