This set$\{E\subset(0,1]:E\text{ is countable or } E^c\text{ is countalbe}\}$is a $\sigma$-algebra.I strongly believe it can't be generated by some countable collection of sets. But I don't know how to prove it rigorously. Thanks!
By Nate Eldredge's hints, I write the details here:
For hint 1: for the each set $A_i\in \mathcal{A}$ with $A_i^c$ is countable, we can replace $A_i$ by $A_i^c$. For the new set $\mathcal A'$, we have $\sigma(\mathcal A)=\sigma(\mathcal A')$. So "without loss of generality, we can assume all the $A_i$ are countable."
For hint 2: Because every $A_i\in \mathcal{A}$ is countable, so $\mathcal A$ can only cover countable many point in $(0,1]$. There must be some point $x\in (0,1]$, and $x\notin A_i,i=1,2,...$. Let $B=\{x\}$.
For hint 3: Let $A=\bigcup_{i=1}^\infty A_i$, and $\mathcal F=\{E\subset(0,1]:E\subset A\text{ or }E^c\subset A\}$ We can easily check $\mathcal F$ is a $\sigma$-algebra, and $\mathcal A\subset\mathcal F$. Hence $\sigma(\mathcal A)\subset\mathcal F$ but $B=\{x\}\notin\mathcal F$.