Externally just keyboard and mouse. Internally just the DVD drive and the SSD running Windows 7. The original secondary hard drive (D: drive) has been unplugged internally since the first couple of failed upgrade attempts.
This system runs an ASRock X79 Extreme3 motherboard, which was quite a complicated high-end motherboard back when this system was built. Various additional third-party controller chips were added to the board, like the ASMedia ASM1061 SATA3 6.0 Gb/s controller, and Etron EJ168A USB 3.1 controller.
I can't help but wonder if this problem is something to do with the ASMedia ASM1061 SATA3 controller.