  The purpose of this short problem paper is to raise an extremal question on
set systems which seems to be natural and appealing. Our question is: which set
systems of a given size maximise the number of $(n+1)$-element chains in the
power set $\mathcal{P}(\{1,2,\dots,n\})$? We will show that for each fixed
$\alpha>0$ there is a family of $\alpha 2^n$ sets containing $(\alpha+o(1))n!$
such chains, and that this is asymptotically best possible. For smaller set
systems we are unable to answer the question. We conjecture that a `tower of
cubes' construction is extremal. We finish by mentioning briefly a connection
to an extremal problem on posets and a variant of our question for the grid
graph.
