Prove that $L$ is closed under Kleene star iff $L=NL$
Let $A$ be the language consisting of the following words:
$$
(u,v)|\Sigma^*|(v,w),
$$
where $u,v,w$ are numbers encoded in binary. This language is in $\mathsf{L}$.

Given a directed graph $G$, we can encode it as a list of edges in the following way:
$$
(x_1,y_1)(x_1,y_1)|(x_2,y_2)(x_2,y_2)|\cdots|(x_m,y_m)(x_m,y_m)
$$
where $x_i,y_i$ are vertex numbers encoded in binary. Furthermore, this string, which we denote by $\langle G \rangle$, can be output by a logspace machine.

Suppose now that we are given two vertices $s,t$, and that there is an $s$-$t$ path of length $\ell$ of the form $(s,x),\ldots,(y,t)$ in $G$. Then I claim that for all $L \geq \ell$,
$$
(s,x)|\langle G \rangle^L|(y,t) \in A^*.
$$
Conversely, if such a string belongs to $A^*$ then there is an $s$-$t$ path in $G$. In other words,
$$
\exists \text{$s$-$t$ path in $G$} \longleftrightarrow (s,t) \in G \lor \exists x,y \text{ s.t. } (s,x)|\langle G \rangle^n|(y,t) \in A^*.
$$

If we assume that $\mathsf{L}$ is closed under Kleene star then $A^* \in \mathsf{L}$, and so we can evaluate the right-hand side formula in logspace. Since $s$-$t$ reachability is $\mathsf{NL}$-complete, we deduce that $\mathsf{L}=\mathsf{NL}$.