  We determine the asymptotic behavior of the entropy of full coverings of a $L
\times M$ square lattice by rods of size $k\times 1$ and $1\times k$, in the
limit of large $k$. We show that full coverage is possible only if at least one
of $L$ and $M$ is a multiple of $k$, and that all allowed configurations can be
reached from a standard configuration of all rods being parallel, using only
basic flip moves that replace a $k \times k$ square of parallel horizontal rods
by vertical rods, and vice versa. In the limit of large $k$, we show that the
entropy per site $S_2(k)$ tends to $ A k^{-2} \ln k$, with $A=1$. We
conjecture, based on a perturbative series expansion, that this large-$k$
behavior of entropy per site is super-universal and continues to hold on all
$d$-dimensional hyper-cubic lattices, with $d \geq 2$.
