As beginner in CS, I also became interested in CFG descriptional complexity. First of all I considered its variables complexity and got the following results for the particular language $L = (ab)^* + (ba)^*$:
- There exists CFG with two variables which generates $L$
- There exist no CFG with one variable which generates $L$
For proving (1) we may just consider the following CFG $$ S \rightarrow bAa \mid A , \, A \rightarrow abA \mid \varepsilon$$
The second proposition is a bit more tricky. Suppose that there exists CFG $G = (\{S\}, \{a, b\}, P, S)$ such that $L = L(G)$. Every production is $S \rightarrow \alpha$ for some $\alpha \in \{a, b, S\}^*$. Say, $\alpha$ contains $k$ instances of $S$; for every $u_1, \ldots u_k$ define the $\widetilde{\alpha}(u_1, \ldots u_k) \in \{a, b\}^*$ to be the word obtained from $\alpha$ by substituting $u_1$ into the first instance of $S$, $u_2$ into the second and so on. Then $u_1, \ldots u_k \in L$ implies $\widetilde{\alpha}(u_1, \ldots u_k) \in L$: as $L$ is generated by $G$, one can derive $\widetilde{\alpha}(u_1, \ldots u_k) \in L$ by using $S \rightarrow \alpha$ and then deriving every $u_i$ from $S$'s. By using that we can prove that in right-hand side of every production $S$ can't be followed immediately by any symbol:
- if $S$ is followed by $a$, we have subword $Sa$ in $\alpha$, then we can derive $ba$ from $S$, thus we can derive the word which contains two $a$'s in the row, which obviously doesn't lie in $L$;
- if $S$ is followed by $b$, having subsword $Sb$ of $\alpha$ we can derive $ab$ from $S$ and thus obtain the word which contains two $b$'s in the row;
- if we have subword $SS$ of $\alpha$, we can derive $ab$ from the first $S$ and $ba$ from the second.
Also $S$ can't follow neither letter nor $S$ in right-hand side of every production. So $G$ can only have productions of sort $S \rightarrow S$ and $S \rightarrow w$ for $w \in L$. Thus $L(G)$ is finite, which is contradiction.
You can also look to the Grushka's paper: https://www.sciencedirect.com/science/article/pii/S0019995871905195