Primitive, non-2-transitive groups with very large orbitals?

Question

Let $G$ be a transitive permutation group on a set $X$ with $n$ elements. Assume that $G$ is primitive, i.e., $G$ preserves no non-trivial partition of $X$. Assume as well that $G$ is not $2$-transitive, i.e., the action of $G$ on pairs of distinct elements of $X$ is not transitive. Because $G$ is not $2$-transitive, the action of a pointwise stabilizer $G_x$ ($x\in X$) on $X\setminus \{x\}$ cannot be transitive.

Question: how large can the largest orbit of $G_x$ be? Is it possible that it be of size $\geq 0.99n$, say?

(If $G$ is not required to be primitive, then $G_x$ can have very large orbits. For example, for $n$ even, let $G$ be generated by transpositions $(2m-1\;\; 2m)$ and by elements of $\text{Sym}(n/2)$ acting on the $n/2$ possible pairs $\{2m-1,2m\}$ of elements of $\{1,2,\dotsc,n\}$. Then $G$ is not $2$-transitive, but $G_x$ has $\{3,4,\dotsc,n\}$ as an orbit.)

Answer 1

Let $G=S_n$, acting on the ${{n} \choose {2}}$ $2$-sets from $[n]$. The stabilizer of a $2$-set $X$ is maximal in $S_n$ and has two orbits on the remaining $2$-sets, one consisting of those that intersect $X$ nontrivially. The other orbit has size ${{n-2} \choose {2}}$. Now let $n \rightarrow \infty$.