It surprises me that there's a unique 12 digit answer to this. If you bet proportion f of your income every round, then if you win n coin flips, shouldn't you have (1+2f)^n (1-f)^(1000-n) £? For a given f, it takes a certain number of heads for this value to exceed 1B£, but I'd expect a range of f values that all require the same minimal number of heads results to break 1B£. Unless there's a value of f that produces exactly 1B£ for the minimal needed number of heads (or the set of such f differ only after 12 digits). That seems unlikely...
EDIT: Just realized that the problem asks for the odds of being a billionaire, not f. So there are probably a good number of f that work, but they all produce the same odds.
EDIT: Just realized that the problem asks for the odds of being a billionaire, not f. So there are probably a good number of f that work, but they all produce the same odds.