jmc

First, we need to find $L_{10}$. We find that $$\begin{array}{rl}{L}_2& =L_1+L_0=3,\\ L_3& =L_2+L_1=4,\\ L_4& =7,\\ L_5& =11,\\ L_6& =18,\\ L_7& =29,\\ L_8& =47,\\ L_9& =76,\\ L_{10}& =123\end{array}$$Thus, $L_{L_{10}}=L_{123}$. To find its units digit, we continue listing more values in the sequence until we reach a pattern: the units digit of $L_{11}$ is that of $123+76$, and so is $9$; that of $L_{12}$ is $2$; and that of $L_{13}$ is $1$. Hence, the units digit repeats starting from here, with a period of $12$. As $123=12\times 10+3$, then the units digit of $L_{123}$ is the same as that of $L_3$, or $\boxed{4}$.