One very interesting cube-free infinite binary word is the Kolakoski word (also known as the Kolakoski sequence). The sequence is named after William Kolakoski, who introduced it in a problem published in the American Mathematical Monthly in 1965.
It seems that it was recently (last year?) discovered that the sequence was published earlier, in 1939, by Rufus Oldenburger in his paper Exponent trajectories in symbolic dynamics (Transactions of the American Mathematical Society, vol. 46, pp. 453-466).
One currently open problem (see, for example, problem 10 here) is to prove (or disprove) that the limiting frequency of each of the two characters exists and is equal to 1/2. Perhaps some insight can be found in Oldenburger’s paper?