At a high level I’d probably try and exploit the fact that every bit sequence with a given number of H and T has equal probability. e.g., HHHT HHTH HTHH THHH are equally probable and so can be mapped to four different values. That still only gets me 2 bits (50%) but other combinations (e.g., variations on HHTT) could get me log2(6) bits. I’m guessing with a higher number of flips I could extract (on average) more and more as a proportion. No clue what the asymptote would be.
