There are a LOT of variables that come into play here. The best answer is, what feels and works best for you. The tire's load range, the tire's sidewall stiffness, the weight of your vehicle, the terrain you are on, the issue of being able to air back up after a trail run before hitting the pavement, beadlocks, etc. If you have a way to air back up at the end of the day, it is easier to play around with to get the best PSI. If not, I wouldn't air down much below about 18 psi, because you will have a lot of sidewall roll once you get back onto pavement.
Personally, I run 35x12.50x15 Goodyear MTRs on non-beadlock wheels. If I'm running logging roads and I just want a little less harshness in the ride, I drop the pressure to about 18 psi. Rocks, I'll drop it down to about 12 psi. I could go lower, maybe down to 10, but I'm always worried about popping the bead without beadlocks. However, my MTR's seem quite a bit stiffer than the worn out KM's that I had before. I just did the Rubicon at 12 psi and didn't have a problem.
Finally, I have a VIAIR compressor and a 2.5 gallon tank for re-inflation. The tank does basically nothing for tire re-inflation and I wouldn't add it again. I think it actually hinders re-inflation, since it is trying to fill the tank and the tire at the same time. It is nice for my air lockers, though, since the compressor doesn't run nearly as much when turning on and off the lockers a bunch.
