  Assign to each site of the integer lattice $\Zd$ a real score, sampled
according to the same distribution $F$, independently of the choices made at
all other sites. A lattice animal is a finite connected set of sites, with its
weight being the sum of the scores at its sites. Let $N_n$ be the maximal
weight of those lattice animals of size $n$ that contain the origin. Denote by
$N$ the almost sure finite constant limit of $n^{-1} N_n$, which exists under a
mild condition on the positive tail of $F$. We study certain geometrical
aspects of the lattice animal with maximal weight among those contained in an
$n$-box where $n$ is large, both in the supercritical phase where $N > 0$, and
in the critical case where $N = 0$.
