Infinite Union of non-regular languages
Let $L$ be a non-regular language over $\Sigma$, and let $w_1,w_2,\ldots$ be an enumeration of $\Sigma^*\setminus L$. Define $L_i = L \cup \{ w_j : j \geq i \}$. Each $L_i$ is non-regular (why?), $L_i \subset L_{i+1}$, and $\bigcup_i L_i = \Sigma^*$.