  A word is square-free if it does not contain any square (a word of the form
$XX$), and is extremal square-free if it cannot be extended to a new
square-free word by inserting a single letter at any position. Grytczuk,
Kordulewski, and Niewiadomski proved that there exist infinitely many ternary
extremal square-free words. We establish that there are no extremal square-free
words over any alphabet of size at least 17.
