Let $x_1, x_2, x_3, \ldots$ be the sequence $0, 1, 4, 6, 8, 9, \ldots$ of non-prime, non-negative integers.
Let $a^{x_i}$ be a string of $x_i$ number of $a$’s (that is, $a^{x_3} = a^4 = aaaa$)
Let the language $L_i = a^* - \left \{a^{x_i} \right \}$.
Now consider $L =$ The infinite intersection of the sequence of languages $L_1, L_2, \ldots$. That is, $$L = \bigcap_{i=1}^{\infty}L_i = L_1 \cap L_2 \cap L_3 \cap \ldots$$
Note that $L = \left \{a^p \mid p \text{ is prime}\right \}$.
Hence L is not regular.
Another easy example would be (this one with unions as opposed to intersections):
Let $L_i = \left \{a^i b^i \right \}$ for some given value $i$.
Then, $L_1 = \{ab\}, L_2 = \{aabb\}, L_3 = \{aaabbb\}$ and so on.
Now consider $L =$ The infinite union of the sequence of languages $L_1, L_2, \ldots$ That is, $$L = \bigcup_{i=1}^{\infty}L_i = L_1 \cup L_2 \cup L_3 \cup \ldots$$
Thus, $L = \left \{ a^i b^i \mid \forall i > 0 \right \}$ which is CFL but not Regular.