Hmmm, you are entirely correct. I ran the numbers as log (150,000) / log (2), and got 17. This was on the assumption that Alcor probably doesn't handle more than a single patient in a day.

More conservatively, I should have probably assumed Alcor handles ~10/year (based on 2010 figures). At that point we get 0.025 per day, which is about 5 additional doublings. So it looks like a fairer number would have been 22 doublings, assuming that this rule holds true. Thus, my padding to triple was probably slightly pessimistic, given the assumptions I made.