Great! I was fearing you were in the gambler's fallacy :-)
May I ask you why you are so interested in the busting probability of funny martingales?
You can ask, but I don't know what to tell you.
I'm interested in mathematics, and this seems like it should have a neat solution.
I don't know that you can call it a "martingale" by the way, since it doesn't reset on a win.
If you are interested in martingale / gamblers fallacy / hothand (as I am) Its worth looking at the Fibonacci variants of Martingale. I havnt read all 216 pages of this thread but I did do a search for the term Fibonacci and nothing came up. For example:
Start with the Fibonacci series: 1,1,2,3,5,8,13,21 .... starting with 1:
a) Place 50/50 bet
b) If win, increase 1 step
c) If lose, go back 2 steps
The psychology is that you lose only 1 unit 50% of the time, but the other 50% of the time the sky is the limit! So the "bust" in this case is not losing your pot in a streak of bad luck, but rather, having a streak of good luck and hitting the just/dice limit.
From an entertainment point of view, i believe you can play fibonacci much longer then classic Martingale (assuming 1 unit as the initial bet). I dont think it changes the houses expected value, aside from possibly increasing the propensity for he user to play again through positive reinforcement.
Just a few years ago a psychologist (Daniel Kahaneman) won the nobel prize in economics for related work in behavioral economics. ANyhow.... great discussion, and I think the math behind the JD website is very very cool!
(Fibonacci caveat - Strictly speaking, you want to start with 1 unit, then multiply repeatedly by Phi [~1.68=sqrt(5)+1)/2] which quickly converges to the more popular integer series. Phi is one of those cool constants related to DaVinci, Pi, pythagoras and that 3x3 grid on your iphone camara. )