  An irredundant base of a group $G$ acting faithfully on a finite set $\Gamma$
is a sequence of points in $\Gamma$ that produces a strictly descending chain
of pointwise stabiliser subgroups in $G$, terminating at the trivial subgroup.
Suppose that $G$ is $\operatorname{S}_n$ or $\operatorname{A}_n$ acting
primitively on $\Gamma$, and that the point stabiliser is primitive in its
natural action on $n$ points. We prove that the maximum size of an irredundant
base of $G$ is $O\left(\sqrt{n}\right)$, and in most cases $O\left((\log
n)^2\right)$. We also show that these bounds are best possible.
