On some BKs (eg. Ashford), you can hear the thermostat click shut. If you do this on a cold stove, then we know your minimum setting must be somewhere above this, but how far above may vary with chimney height and other factors.
Ashful's 'other factors' sound mysterious because it is actually more complex than you'd think. You will never be able to take a sharpie and mark "minimum" on your thermostat, but you will get a feel for where it is.
The lowest setting will vary from load to load. It isn't a fixed setting because of many factors. It'd be pretty complicated to to wangle it all into a formula, but you get a pretty good feel for it all (especially since a lot of the factors stay reasonably constant day to day sometimes).
If you asked me to set my stove as low as possible without stalling the cat, I'd look at the outside temperature (higher temps mean worse draft), the starting bed of coals (if there isn't one or it is very low, I am gonna have to run hotter, much hotter if the wood's less dry), the dryness of the wood (wetter wood is harder to burn and puts out less heat), the size of the splits (more surface area means more combustion, and lots of tiny splits seem to be able to form a raging inferno even with the air waaay down), how much time I have to babysit the stove (I know that I can turn the air down right away on dry pine, but oak needs a minute of it doesn't have a good coal bed; if either one is wet, you might be in for a lot of babysitting if you don't have an excellent coal bed.)
I'm sure that I'm missing something in there, but it's not coming to mind...
Edit: Missed one of the most important ones for my insert (but this is not nearly as important for a freestanding stove): House temp and weather forecast: If the house starts cool and the temp is dropping, I will be turning the fan on... And now my minimum thermostat setting needs to be higher!
Anyway, that looks like a lot of stuff, but it's a lot more intuitive than is might sound... Just use the stove and you'll get the hang of it.
If it sounds intimidating, know that most people use these stoves without ever knowing about or considering most of (or any of) that stuff, and they get their houses heated too.