  A word is \emph{square-free} if it does not contain non-empty factors of the
form $XX$. In 1906 Thue proved that there exist arbitrarily long square-free
words over $3$-letter alphabet. We consider a new type of square-free words. A
square-free word is \emph{extremal} if it cannot be extended to a new
square-free word by inserting a single letter on arbitrary position. We prove
that there exist infinitely many extremal words over $3$-letter alphabet. Some
parts of our construction relies on computer verifications. We also pose some
related open problems.
