I'm interested in right-infinite words over a two-letter alphabet that do not contain subwords of the form $xxx$. (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?

Added

From the comments it seems that I need to clarify the terms I use. 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 $0110$. A word is cube-free if it does not contain subwords of the form $xxx$ where $x$ is a word of one or more letters.