In an MO question here @IosifPinelis shows that the ratio of expectations $\mathbb{E}(A)/\mathbb{E}(B)$ of the largest (say $A$) and smallest (say $B$) gap resulting from $n$ uniform random variables on $(0,1]$ tends to infinity as $n\rightarrow \infty.$
In an earlier question, linked in the above question, he also showed that $\mathbb{E}(A/B)$ equals to infinity for all $n>2.$
I have a related question (thanks @YuvalPeres for your comments):
Let $G_{(1)}$ be the smallest, $G_{(2)}$ be the second smallest, etc. and let $G_{(n)}$ be the largest gap.
What is the fastest growing sequence $\ell(n)$ such that $$\lim_{n\rightarrow \infty} \frac{\mathbb{E} G_{(\ell(n))}}{\mathbb{E} G_{(1)}}<\infty?$$
Edit: Simulations seem to confirm the proof by @MattF. that $\ell(n)=2$ yields $7/3.$