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(I don't really think that I am, though, because by definition, you WILL win, and when you do, your BR will be larger than when it started.)

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Well, that argument meets to its complement, which is: by definition, you WILL lose, and lose many many many times in a row on occasion. And the longer you play, you will set new records for longer and longer losing streaks.

The accumulated small wins you are thinking of are balanced by an extreme theoretical tail or far-out reach of the matrix of possible paths, on the negative side.

The losses compound geometrically more and more, the rarer (longer) they are.

Also, although this is tangential and more complicated, if you had an infinite number of gamblers playing this system, might you have at least one gambler who would never get to make a single winning bet? So he might never complete the first series?