A word $y$ is a subword of $w$ if there exist words $x$ and $z$ (possibly empty) such that $w=xyz$. Thus, $01$ is a subword of $0110$, but $00$ is not a subword of $0110$. I'm interested in right-infinite words over a two-letter alphabet that do not contain subwords of the form $xxx$, where $x$ is a word of one or more letters. (For example, the Thue-Morse word, the Kolakoski word, Stewart's choral sequence, and so on.) In particular, I would like to know if there are any general statements about *all* such words. For example, are there an infinite number of them? Is there any way to classify them?

It's possible that one way to classify cube-free infinite binary words is to group them according to their subwords. For example, the Kolakoski word has the subword $00100$ whereas the Thue-Morse word does not, so they belong in different classes. The words created in Tony's answer (see below) have the same subwords (the Thue-Morse word is recurrent), so they belong to one class. I suppose there an infinite number of these classes.

Another possible way to classify cfib words is to group them according to their subword complexity. For example, Stewart's choral sequence has a subword complexity of $2n$ (where $n$ is the length of the subword), so we can group it with other cfib words with subword complexity $2n$. Is the subword complexity of the Kolakoski word known?