  Fici, Restivo, Silva, and Zamboni define a $\textit{$k$-anti-power}$ to be a
concatenation of $k$ consecutive words that are pairwise distinct and have the
same length. They ask for the maximum $k$ such that every aperiodic recurrent
word must contain a $k$-anti-power, and they prove that this maximum must be 3,
4, or 5. We resolve this question by demonstrating that the maximum is 5. We
also conjecture that if $W$ is a reasonably nice aperiodic morphic word, then
there is some constant $C = C(W)$ such that for all $i,k\geq 1$, $W$ contains a
$k$-anti-power with blocks of length at most $Ck$ beginning at its
$i^\text{th}$ position. We settle this conjecture for binary words that are
generated by a uniform morphism, characterizing the small exceptional set of
words for which such a constant cannot be found. This generalizes recent
results of the second author, Gaetz, and Narayanan that have been proven for
the Thue-Morse word, which also show that such a linear bound is the best one
can hope for in general.
