May 2015
There are three solutions (up to permutations of the amounts) for the money the three players start with, so that at least 12 rounds (or one hour) of betting will take place with amounts no greater than 255: (175,199,223), (197,205,213), and (209,217,225). Incidentally, there is no solution for 13 rounds of play.
Thanks to John Tromp for creating the relevant sequence: https://oeis.org/search?q=A256001.
Goran Zuzic uploaded a proof (https://gist.github.com/Zuza/36b5ea4d199d14d8f22c) that for any three numbers, the game can end in finite amount of time.