Lower and upper bound for order of subgroup generated by a subset

Question

Let $G$ be a finite group and $A\subseteq G$.

Is there any (non-trivial and useful) lower bound for $\lvert\langle A \rangle\rvert$ (order of the subgroup generated by $A$)? What about upper bound?

Answer 1

All the finite simple groups are $2$-generated, so I guess the answer is no for the upper bound.

Addendum See for instance this survey paper by Aner Shalev, in which he mentions (p. 386, following Theorem 5) that if $G$ is a finite simple groups, and $x \in G$ is an arbitrary element, different from the identity, then there is $y \in G$ such that $\langle x, y \rangle = G$.