Regularity of infinite concatenation
The answer is negative. Let $L_n = \{\epsilon, a^{3^n}\}$ for $n \geq 0$, and let $L_n = \{\epsilon\}$ for $n &lt; 0$. The infinite concatenation of the collection $\{ L_n : n \in \mathbb{Z} \}$ is
$$
L = \{ a^m : \text{$m$ is the sum of distinct powers of $3$} \}.
$$
Since $L$ is unary, in order to show that $L$ is not regular, we need to show that the set $S = \{ m : \text{$m$ is the sum of distinct powers of $3$} \}$ is not eventually periodic. We can see this in many ways. For example, $S$ is infinite but has density $0$ in the limit.