Well, take a moment to reflect. Is a word or a string one of the most important and most common entities in this world? What are the most basic properties of a word or string?
- its alphabet
- its length
- its sound
- its meaning
- its syntax or grammar
- its origin
- its usage as how to form sentences.
- hmm, does it repeat part of itself? Or never (square-free)?
It looks like the property of being square-free is almost a fundamental property of a word (or a phrase, or a sentence).
In fact, it is amazing to me that the formal study of square-free words has not made it way into most of textbooks about the theory of automata and computation. It appears that the study of square-free words turns out to be a rather isolated field with less impact.
Nevertheless, the study of square-free words and related concepts is interesting and fruitful, as you have observed. I am, for one, charmed by the square-free words of infinite length over three letters 1.
"Real-life" application related to square-free words are sparse indeed. Abelian square-free words in algorithmic music by Laakso, T. might count as one; however, I cannot find that paper. (Gefwert, C., Orponen, P., Seppänen, J. (eds.) Logic, Mathematics and the Computer, vol. 14, pp. 292–297. Finnish Artificial Intelligence Society, Symposiosarja, Hakapaino, Helsinki).
Here are two related exercise to entertain your mind.
An abelian square is a subsequence of the form $s_1s_2$ where $s_2$ is a permutation of $s_1$. For example, $abcbac$ is an abelian square since $bac$ is a permutation of $abc$.
Exercise 1. (easy) Let the alphabet consist of 3 letters. Show a word of length 7 that is free of abelian square. Show that every word of length eight contains an abelian square.
Exercise 2. (very very hard) There is an infinite abelian-square-free word over an alphabet of size four.