Consider a set of simple closed curves on a surface of genus $g$ which fill
the surface and which pairwise intersect at most once. We show that the
asymptotic growth rate of the smallest number in such a set is $2\sqrt{g}$ as
$g \to \infty$. More generally, we give a precise asymptotic for filling sets
of curves which pairwise intersect at most $K \geq 1$ times. We then bound from
below the cardinality of a filling set of {\it systoles} by
$\frac{g}{\log(g)}$. The topological condition that a set of curves pairwise
intersect at most once is thus quite far from the geometric condition that a
set of curves can arise as systoles.