Hamsfarthing: Seems that despite the GOG version of the game not needing the original CDs in any way, it still does some sort of check for an optical drive.
The only solution I've been able to come up with is installing Daemon Tools or something similar, as that creates a virtual optical drive which is sufficient for the game to load up normally.
I can confirm this. I had the same problem, then
I plugged in my external DVD drive and EoC stopped asking for a disc.
Works like charm now. I have something to play while waiting for
Elite: Dangerous!
Thank you Hamsfarthing and others!