When you add all the countable unions and complements the class of sets you get is not closed under countable unions and complements.
Given $F$, a family of subsets of $X$, let's say $C(F)$ is $F$, plus $\emptyset$, $X$, and all countable unions and complements of elements of $F$. Now say $A_1,A_2,\dots\in F$. Then $B=\bigcup A_n\in C(F)$, but the complement of $B$ need not be in $C(F)$. (Similarly, the union of the complements of $A_n$ need not be in $C(F)$.
You could try just repeating it, considering $C(C(F))$. That's not closed under countable unions and complements either. Give up, do it infinitely many times, and take the union of $C(F)$, $(C(C(F))\dots$? That's not enough either, because if $A_n\in C^n(F)$ we never picked up the union of the $A_n$ at any stage.
So apply $C$ infinitely many times, take the union, and then apply $C$ one more time? Not enough. Speaking informally, you have to apply $C$ uncountably many times before you get something that's guaranteed to be closed under countable unions and complements. You can make sense of "apply $C$ uncountably many times" via transfinite recursion; you need to apply $C$ exactly $\omega_1$ times, where $\omega_1$ is the first uncountable ordinal.
(Transfinite recursion is what it actually is - if you look for an explanation of this online you may have better luck searching for "tranfinite induction"...)