  We investigate the ascending Loewy socle series of Leavitt path algebras
$L_K(E)$ for an arbitrary graph $E$ and field $K$. We classify those graphs $E$
for which $L_K(E)=S_{\lambda}$ for some element $S_{\lambda}$ of the Loewy
socle series. We then show that for any ordinal $\lambda$ there exists a graph
$E$ so that the Loewy length of $L_K(E)$ is $\lambda$. Moreover, $\lambda \leq
\omega $ (the first infinite ordinal) if $E$ is a row-finite graph.
