In a comment made by Gjergji Zaimi to this older question, it is conjectured that $\frac 73$ is the threshold separating countability and uncountability of the sets of infinite binary words having a given critical exponent. In a later comment, James Currie cites the (remarkable!) paper:
"J. Karhumaki and J. Shallit. Polynomial versus exponential growth in repetition-free binary words. J. Combin. Theory. Ser. A 105 (2004), 335–347."
as the one proving the conjecture. Is it so, though?
In general, it is not true that, if every element of $S\subset\{0,1\}^\omega$ has at most polynomially many finite subwords of length $n$, then $|S|\le \aleph_0$. In fact, it is sufficient to allow $n+1$ subwords of length $n$ to get an uncountable set, as exemplified by the well-known Sturmian words. So the conjecture ie either false or there must be a more specific argument involving critical exponents to establish the conjecture. What is it?