Regular language? Union of the dictionary and an non-regular language
The best way to see that $L_1 \cup L_2$ is not regular is through closure properties of the regular languages.

Theorem. If $L_1$ is finite and $L_2$ is not regular, then $L_1 \cup L_2$ is not regular.

Proof. Since $L_1 \setminus L_2$ is finite, it is regular.
Suppose $L = L_1 \cup L_2$ were regular. Then $L \setminus (L_1 \setminus L_2) = L_2$ would also be regular, contrary to assumption. $\qquad\square$