  The hypergraph regularity lemma -- the extension of Szemer\'edi's graph
regularity lemma to the setting of $k$-uniform hypergraphs -- is one of the
most celebrated combinatorial results obtained in the past decade. By now there
are several (very different) proofs of this lemma, obtained by Gowers, by
Nagle-R\"odl-Schacht-Skokan and by Tao. Unfortunately, what all these proofs
have in common is that they yield regular partitions whose order is given by
the $k$-th Ackermann function.
  In a recent paper we have shown that these bounds are unavoidable for
$3$-uniform hypergraphs. In this paper we extend this result by showing that
such Ackermann-type bounds are unavoidable for every $k \ge 2$, thus confirming
a prediction of Tao.
