Let $\{X_v, v \in \mathbb{Z}^d\}$ be i.i.d. random variables, and
$S(\xi) = \sum_{v \in \xi} X_v$ be the weight of a lattice animal $\xi$. Let
$N_n = \max\{S(\xi) : |\xi| = n$ \text{and $\xi$ contains the origin}\}$ and
$G_n = \max\{S(\xi) : \xi \subseteq [-n,n]^d\}$ . We show that, regardless of
the negative tail of the distribution of $X_v$ , if $\mathbf{E}( X_v^+)^d
(\log^+ X_v^+))^{d+a} < + \infty$ for some $a>0$, then first, $\lim_n
n^{-1} N_n = N exists, is finite and constant a.e.; and, second, there is a
transition in the asymptotic behavior of $G_n$ depending on the sign of $N$: if
$N > 0$ then $G_n \approx n^d$, and if $N < 0$ then $G_n \le cn$, for
some $c > 0$. The exact behavior of $G_n$ in this last case depends on the
positive tail of the distribution of $X_v$; we show that if it is nontrivial
and has exponential moments, then $G_n \approx \log n$, with a transition from
$G_n \approx n^d$ occurring in general not as predicted by large deviations
estimates. Finally, if $x^d(1 - F(x)) \to \infty$as $x \to \infty$, then no
transition takes place.