For a 1D particle in a box, energy levels are exactly proportional to $n^2$.
For the harmonic oscillator, $E_n\sim n$. And for a particle in an $|x|^\alpha$ potential, the energies are $\sim n^\beta$ with $\beta <2$ from the WKB approximation.
I also vaguely recall that the eigenvalues for a regular Sturm-Liouville problem are $\sim n^2$.
All of this suggests that the eigenvalues cannot rise faster than $n^2$. Is my intuition correct here?