Is the union of two non-regular context-free languages always non-regular?
Let $L$ be any non-regular context-free language over $\{0,1\}$, for example $\{ 0^n 1^n : n \geq 0 \}$. Take
$$
\begin{align*}
L_1 &amp;= 0L \cup 1\Sigma^* \cup \{\epsilon\}, \\
L_2 &amp;= 1L \cup 0\Sigma^* \cup \{\epsilon\}.
\end{align*}
$$
Then $L_1$ and $L_2$ are also non-regular context-free (exercise), but $L_1 \cup L_2 = \Sigma^*$.