What is

$$ \limsup_{n\to\infty} \sup_{\deg(G)=n} \left( \frac{\max_{x\in G} \left|\operatorname{conj}(x)\right|}{\operatorname{ord}(G)}\right),$$

$G$ transitive permutation group?

And what are the examples of finite permutation groups containing a large conjugacy class?

$\deg(G)$ resp. $\operatorname{ord}(G) $ are degree and order of $G$, $\operatorname{conj}(x)$ is the conjugacy class of $ x\in G $