I believe that in the limit as number of steps approaches infinity, the amount risked approaches zero, your expected losses approach zero, and so your chance of winning approaches your chance of winning with a single bet on a zero house edge game.
But let's keep that quiet, eh?
I doubt it. Imagine the house edge was 90% instead of 1%. Would you thus be able
to turn an arbitrarily unfair game into a fair one?
I guess the limit depends on the initial house edge and is strictly positive.
If someone wants to work it out, better first find out the optimal proportion
to break your capital into two bets. Then extend this to n bets; then take the limit n to infinity.
It may be a nice exercise in calculus.
by the way, while doing one martingale sequence is better than doing a single bet, doing
many martingale sequences is strictly worse than doing one. And this - "grinding the martingale" -
is what our players seem to most enjoy doing.
Of course, saying "it's more rational to bet just once" is just as useless as saying "it's more rational
not to bet at all," since it doesn't take into account the "fun" part.
A sad fact: more rational = less fun.