  An overlap-free (or $\beta$-free) word $w$ over a fixed alphabet $\Sigma$ is
extremal if every word obtained from $w$ by inserting a single letter from
$\Sigma$ at any position contains an overlap (or a factor of exponent at least
$\beta$, respectively). We find all lengths which admit an extremal
overlap-free binary word. For every extended real number $\beta$ such that
$2^+\leq\beta\leq 8/3$, we show that there are arbitrarily long extremal
$\beta$-free binary words.
