Let's say you lose it all if you ever get 9 losses in a row (1,2,4,8,16,32,64,128,256).  This isn't quite true, because even if your first 9 bets are all losses you still have 0.001 left, and if you've won some sequences before your 9 losses you'll have even more left.

And you double if you have 512 successful sequences before you hit 9 losses in a row.

I assume you're playing the 49.5% 2x game, and so your probability of hitting 9 losses in a row is 0.505**9
So your probability of any single sequence being successful is (1 - 0.505**9).
And your probability of having 512 successful sequences in a row is (1 - 0.505**9) ** 512

That's 0.3345881014449683, or about 1 in 3.

I wonder why we differ so much.