After each pass through the press, I carved away more of the plate, mixed up another ink color, and did it all over again. (Notice how the ink is making the paper more translucent.)
Five layers is the answer to my question. After five layers, the ink is pushed all the way though the paper and the new colors start to be rejected by the print.
It's hard to capture in a photo, but the paper fully saturated with ink is really quite translucent. It glows from within, an effect that can never be achieved with standard cotton watercolor paper or any mechanical printing process.
I have heard this method of printmaking called a "suicide print" because there's no going back. Once the plate is carved, there's no uncarving it. Reduction woodcuts are the original "limited edition" because there can never be more than were printed the first time. The block is destroyed in the creation.