TITLE: Ubiquitous Zimin words 
QUESTION [10 upvotes]: Let $w$ be a word in letters $x_1,...,x_n$. A value of $w$ is any word of the form $w(u_1,...,u_n)$ where $u_1,...,u_n$ are words. For example, $abaaba$ is a value of $x^2$. A word $u$ is called unavoidable if every infinite word in a finite alphabet contains a value of $u$ as a subword. There is a nice characterization of unavoidable words due to Zimin. A word $u$ in $n$ letters is unavoidable if and only if a value of $u$ is a subword of the $n$th Zimin word $Z_n$ defined by induction: $Z_1=x_1$,...,$Z_n=Z_{n-1}x_nZ_{n-1}$, that is $Z_1=x_1, Z_2=x_1x_2x_1, Z_3=x_1x_2x_1x_3x_1x_2x_1,...$. Zimin words appear very often in algebra. For example, if one lists binary expressions of all numbers $1,2,3,...$ and records the numbers of 0s at the end of the numbers plus 1, one will get $12131214121...$ which is the infinite Zimin word. Values of Zimin words also appeared as $m$-sequences in Levitzki's description of Baer radical (see Jacobson's book "Structure of rings") and in the work of Schutzenberger. The Zimin words have obvious fractal structure, so these words could have appeared in other areas of mathematics as well. 
 Question.  Do Zimin words appear in your area of mathematics?
This might be a "big list" question. But I do not know how big the list is, it may be empty. If it turn out to be big, I will make the question "community wiki". 
 Update 1.  Googling 121312141213121 ($=Z_4$) returns 439 results including a discussion at reddit. 
 Update 2.  The most curious among these links is this link to a US patent. It looks like Zimin words up to $Z_5$ at least have been patented before Zimin introduced them.  
 Update 3.  This is needed for the book with the current title "Words and their meaning" which we are writing together with Mikhail Volkov. There we already have four different applications  of Zimin words to different areas of algebra and would like to mention applications outside algebra as well.

REPLY [5 votes]: I am about to go chair a dissertation defence where the candidate uses an endomorphic image of Zimin words (of his invention) to construct various counterexamples to questions in combinatorics of finite and infinite words, mainly to do with counting palindromes in subwords and scattered subwords of finite and infinite words. The image in question are words on a three-letter alphabet of the form w_0=a, w_1=aba, w_{n+1}=w_nc^nw_n, which the candidate Bojan Basic calls highly potential words. This is obviously a related sequence to Zimin words, and the candidate managed to construct several examples and counterexamples using these, including a counterexample to an already published "theorem" by non-unknown authors.
I suppose it qualifies as an application of Zimin words as highly potential ones are clearly homomorphic images of Zimin words and inspired by them. Don't know if it is too late for your book. Counter-question: has anybody seen these highly potential words before?
Petar Markovic