Title: Iterated doubles of the Joker and their realisability,
Abstract: Let $\mathcal{A}(1)^*$ be the subHopf algebra of the mod~$2$ Steenrod algebra
$\mathcal{A}^*$ generated by $\mathrm{Sq}^1$ and $\mathrm{Sq}^2$. The
\emph{Joker} is the cyclic $\mathcal{A}(1)^*$-module
$\mathcal{A}(1)^*/\mathcal{A}(1)^*\{\mathrm{Sq}^3\}$ which plays a special
rôle in the study of $\mathcal{A}(1)^*$-modules. We discuss realisations of
the Joker both as an $\mathcal{A}^*$-module and as the cohomology of a
spectrum. We also consider analogous $\mathcal{A}(n)^*$-modules for $n\geq2$
and prove realisability results (both stable and unstable) for $n=2,3$ and
non-realisability results for $n\geq4$.