  The largest components of the critical Erd\H{o}s-R\'enyi graph, $G(n,p)$ with
$p=1/n$, have size of order $n^{2/3}$ with high probability. We give detailed
asymptotics for the probability that there is an unusually large component,
i.e. of size $an^{2/3}$ for large $a$. Our results, which extend work of
Pittel, allow $a$ to depend upon $n$ and also hold for a range of values of $p$
around $1/n$. We also provide asymptotics for the distribution of the size of
the component containing a particular vertex.
