I have seen through the years some claims of last day buying and boarding but never saw anyone that could actually substantiate that they did so. That was pre-9/11 and I am almost positive that post-9/11 it is would be even less likely. I would GUESS that about 7 to 5 days out from a sailing would be about the closest you could get only because of boarding manifests that have to be completed by line for immigration and so forth. Documents themselves are not a problem as it is arranged that you pick them up at the pier.
So if you get on board look me up because I would love to meet someone tha actually purchased within days to see how it all went. I am betting that despite openings that the rates would not be just pure giveaway as obviously you must be hoping. But like I said if you get a cabin look me up in 9732, I'll be onboard, and tell me all about it.