Dimensional gap of group representations The problem is inspired by eigenvalue bounds of random Cayley graphs on $SL_2(q)$.
Definition. An infinite series of finite groups $S$ is α-rich if the dimension of the smallest nontrivial representation of $G$ on $\mathbb{C}$ is $\Omega(|G|^\alpha)$ for every $G\in S$.
For example, the series of groups $SL_2(q)$ is $\frac{1}{3}$-rich.
Question. Are there any series of $α$-rich groups with $\alpha>\frac13$?
Known. One only needs to consider simple groups, and by CFSG, it's just going through Lie groups.
Along the lines of David E Speyer, a permutation representation on $O(|G|^\frac13)$ points rules out the possibility for the group to beat $SL_2(q)$.
This holds for all classic groups of type $A_n (n\geq2)$, $B_n(n\geq3)$, $C_n(n\geq2)$, $D_n(n\geq4)$, $^2D_n(n\geq3)$, $^2A_n(n\geq4)$: just consider the action of these groups on the 1-dimensional subspaces of their defining vector space. $^2A_2$ is not in the list, but $^2A_2(q)$ has a representation on $q^2-q+1$ dimensions, hence ruled out.
The same argument rules out $F_4$, $E_n$ and $^2E_6$: All of them are covered in a paper given by Derek Holt. None of them beats $SL_2(q)$. 
$G_2$ and  $^2G_2$ do not beat $SL_2(q)$, by Derek Holt's answer.
For $^2B_2(q)$, see Wikipedia: it almost beats $SL_2$, but it has two characters of dimension $O(q^{3/2})$.
For $^3D_4(q)$, see Deriziotis, D. I., and G. O. Michler. "Character table and blocks of finite simple triality groups $^3D_4(q)$." Transactions of the American Mathematical Society 303.1 (1987): 39-70.: there's a character of dimension $q(q^4-q^2+1)$.
The last case $^2F_4$ is found in Die unipotenten Charaktere fur die GAP-Charaktertafeln der endlichen Gruppen vom Lie-Typ. M. Claßen-Houben; Diplomarbeit, RWTH Aachen; 2005. According to the conventions of the paper, the group has size $q^{52}$, and there's a character of dimension $q^2Φ_{12}Φ_{24}$, which is $O(q^{14})$.
 A: This answer is still not quite complete, but I hope to finish it soon!
The smallest degrees of the faithful permutation representations of the finite simple groups have all been known for a while now. The most convenient reference is probably  Table 4 of this paper, although none of the results are original to that paper. The results for the exceptional groups of Lie type were proved in a series of papers of A. V. Vasil'ev.
Anyway, this approach works for $E_7(q)$ (order about $q^{133}$, minimal permutation degree about $q^{27}$), $E_8(q)$ (order about $q^{248}$, minimal permutation degree about $q^{57}$, $^{2}E_6(q)$ (order about $q^{78}$, minimal permutation degree about $q^{21}$), and ${}^3D_4(q)$ (order about $q^{28}$, minimal permutation degree about $q^9$).
However, the approach  fails for $G_2(q)$, which has order about $q^{14}$ and minimal permutation degree $(q^6-1)/(q-1)$,  $^{2}G_2(q)$ (with $q$ an odd power of $3$), with order about $q^7$ and minimal permutation degree $q^3+1$, and $^{2}F_4(q)$ (with $q$ an odd power of $2$), order about $q^{26}$, minimal permutation degree about $q^{10}$.
So I have been hunting around for results about minimal degrees of representations of $G_2(q)$,$^{2}G_2(q)$, and $^{2}F_4(q)$. For $G_2(q)$ I found them on Page 126 of G. Hiss, Zerlegungszahlen endlicher Gruppen vom Lie-Typ in nicht-definierender Charakteristik, Habilitationsschrift, Aachen 1990. For sufficiently large $q$, these are $q^4+q^2+1$, $q^3+1$, and $q^3-1$ when $q \equiv 0,1,-1 \bmod 3$, respectively. So this is less that $q^{1/3}$, and we are OK.
For $^{2}G_2(q)$, the character tables are computed in this paper. The table is towards the end of the paper, and the smallest character degree is $q^2-q+1$, so again we are OK!
I still need to check $^{2}F_4(q)$.
Added later: i have now located a better reference for minimal degrees of characters of exceptional groups of Lie type, namely Lübeck, Frank, Smallest degrees of representations of
exceptional groups of Lie type. Comm. Algebra 29 (2001),
no. 5, 2147–2169, available here.
In particular, for sufficiently large $q$, the smallest character degree of ${}^2F_4(q^2)$ has degree $11$ in $q$, which is easily sufficient to prove the required result.
