I'm basically interested in the equivalent of the "guessing game" for regular expressions. (See eg https://stackoverflow.com/questions/5440688/the-guess-the-nu... for a rational number solution.)

With a fixed guesser, that would encode all regular expressions / finite automata as sequences of binary digits. (But in a interestingly different way from just serializing the table for a DFA, or writing down the regular expression in ASCII characters.)