How much of the human body can be removed and still result in a living person? What's the minimum?

It depends on the time frame. With a good IV drip going, you could remove nearly all the liver, 1 lung, spleen, pancreas, and a fair deal of the intestines, along with the entire frontal lobe of the brain. However, on an very shortened time frame one could simply live as a head, assuming oxygen is being pumped to the parts of the brain you left intact. Mind you, that the body will start shutting down with such changes and the pain involved along with electrical signalling will be very messed up. The nervous system may just kill the body before a fair amount of this could even be done.