  A word is square-free if it does not contain nonempty factors of the form
$XX$. In 1906 Thue proved that there exist arbitrarily long square-free words
over a $3$-letter alphabet. It was proved recently [7] that among these words
there are infinitely many extremal ones, that is, having a square in every
single-letter extension.
  We study diverse problems concerning extensions of words preserving the
property of avoiding squares. Our main motivation is the conjecture stating
that there are no extremal words over a $4$-letter alphabet. We also
investigate a natural recursive procedure of generating square-free words by a
single-letter right-most extension. We present the results of computer
experiments supporting a supposition that this procedure gives an infinite
square-free word over any alphabet of size at least three.
