My layman reading of this is that this 'appears' to be a violation of martingale theory because this 'should' be a sub-martingale but martingale theory requires finite variance so I suspect the variance of this must go infinite as n tends to infinity?
For 2 flips there are 4 equiprobable cases with gains +125%, -10%, -10%, -64% and the expectation is 10.25%. The expected gain for n flips is 1.05^n-1
If you earn 10% and then lose 10% you are worse off than before. Start with 10: (11-1.1 < 10).
I agree that expectation may not be the most meaningful number here, in the long run is almost sure that we lose everything but there is a zero probability of infinite gains that makes up for it!
Why do you say almost sure to lose everything?
In fact, an even stronger statement is possible: If you start with $1MM, eventually you will fall below $1 and never go above $1 again no matter how long you keep playing.
And all of that not just on a fraction of sequences, but with probability 1.
It's just a regular sub-martingale with 'positive' expectation as you say.