Let’s for any bijection $f: A \to A$ define its support as $$supp(f) = \{a \in A| f(a) \neq a\}$$
Now, let’s define $S_\infty$ as the group of all bijections $\mathbb{N} \to \mathbb{N}$ with finite support. By Cayley Theorem any finite group is isomorphic to a subgroup of $S_\infty$. Therefore, for any finite group $G$ we can define its Cayley length as
$$len_c (G) = \min \{\sum_{\alpha \in A} |supp(\alpha)| | A \subset S_\infty \langle A \rangle \cong G \}$$
Now, we can define a following function:
$$CL(n) = \max \{len_c(G) | |G| \leq n \}$$
What is the asymptotic of $CL$?
I managed to derive the following two bounds:
$$CL(n) = O(n \log(n))$$
This is because any finite group $G$ has a generating set of size $O(\log(n))$ and the size of supports of permutations, corresponding to each of those generators under left multiplicative action is $n$.
$$CL(n) = \Omega(n)$$
Suppose $p$ is prime. Then $len_c(C_p) = p$. Indeed, all non-trivial elements of $C_p$ have order $p$, any permutation of order $p$ has size of support dividing $p$.
However, I do not know, whether any of those bounds is tight...