  We consider the random directed graph $\vec{G}(n,p)$ with vertex set
$\{1,2,\ldots,n\}$ in which each of the $n(n-1)$ possible directed edges is
present independently with probability $p$. We are interested in the strongly
connected components of this directed graph. A phase transition for the
emergence of a giant strongly connected component is known to occur at $p =
1/n$, with critical window $p= 1/n + \lambda n^{-4/3}$ for $\lambda \in
\mathcal{R}$. We show that, within this critical window, the strongly connected
components of $\vec{G}(n,p)$, ranked in decreasing order of size and rescaled
by $n^{-1/3}$, converge in distribution to a sequence
$(\mathcal{C}_1,\mathcal{C}_2,\ldots)$ of finite strongly connected directed
multigraphs with edge lengths which are either 3-regular or loops. The
convergence occurs the sense of an $\ell^1$ sequence metric for which two
directed multigraphs are close if there are compatible isomorphisms between
their vertex and edge sets which roughly preserve the edge-lengths. Our proofs
rely on a depth-first exploration of the graph which enables us to relate the
strongly connected components to a particular spanning forest of the undirected
Erd\H{o}s-R\'enyi random graph $G(n,p)$, whose scaling limit is well
understood. We show that the limiting sequence
$(\mathcal{C}_1,\mathcal{C}_2,\ldots)$ contains only finitely many components
which are not loops. If we ignore the edge lengths, any fixed finite sequence
of 3-regular strongly connected directed multigraphs occurs with positive
probability.
