Let me apologize, for it's not an answer, but merely an extended comment. I like the problem very much. It sounds very classical, and most likely the answer must be known.
Being reformulated, the problem is as follows: if $X$ is an $n$-letter alphabet, we have a rewriting rule
w_1 s s w_2=w_1 s^2 w_2 \to w_1 w_2
where $s$ is nonempty and at least one of $w_1,w_2$ must be nonempty. (In effect, any word that satisfies the condition set by a-boy must be square-free so to speak.) One can also a semigroup with $n$ generators whose relations are determined by the rewriting rule above.
The problem as set asks whether all words over $X$ are reduced to words of certain length $l(n).$ One wonders, however, whether $n=2$ is an exceptional case, and actually there are arbitrary large reduced words. For instance, the following word of length 68 in letters $0,1,2$ is reduced:
For a 4-letter alphabet it is easy to find words of quite large length (I've found by means of computing such words of lenght $\ge 1000.$)