[bzax]

The expected value of this distribution goes up with every iteration, there is no such Kelly point. You could try this with

heads: double your money tails: lose all your money

in which case the expected value is always $1, as you have a 1/2^n chance of having $2^n dollars after n rounds, and 0 otherwise.

The point of discussing ergodicity here, however, is whether you can describe the behavior of the iterated distribution deterministically if you exclude a portion of that distribution which has measure zero.