Let us call the *rank* of a finite group G the number of its irreducible complex representations (up to equivalence). Here is the computation of the order and the rank of the first finite simple groups:  

    gap> it:=SimpleGroupsIterator(10,1000);
    <iterator>
    gap> for G in it do Ch:=CharacterDegrees(G); Print([G,Order(G),Sum(Ch,l->l[2])]); od;
    [ A5, 60, 5 ][ PSL(2,7), 168, 6 ][ A6, 360, 7 ][ PSL(2,8), 504, 9 ][ PSL(2,11), 660, 8 ]  

The points of the following picture corresponds to the $67$ finite simple groups of order less than or equal to $2100000$, where $x$-axis is the rank and $y$-axis the order.
[![enter image description here][1]][1]   
We observe that these points are bounded below by the curve interpolating the points for the groups $\mathrm{PSL}(2,2^n)$, where we get that rank$(\mathrm{PSL(2,2^n)}) = 2^n+1$, for $n \le 7$.   

    gap> for n in [1..7] do Ch:=CharacterDegrees(PSL(2,2^n)); Print([2^n,Sum(Ch,l->l[2])]); od;
    [ 2, 3 ][ 4, 5 ][ 8, 9 ][ 16, 17 ][ 32, 33 ][ 64, 65 ][ 128, 129]

I guess this equality holds in general (let me know in comment). Now $$|\mathrm{PSL(2,2^n)}| = 2^n (2^{2n}-1) = (r-1)((r-1)^2-1)) = (r-2)(r-1)r,$$ with $r = 2^n+1$, which leads to:     

**Question**: Let $G$ be a finite simple group of rank $r$. Is it true that $|G| \ge (r-2)(r-1)r$?     

If so, do you expect the existence of a proof without CFSG?    
(because motivation: extend such a result to the simple integral fusion rings)

    gap> for r in [5..15] do Print([r,(r-2)*(r-1)*r]); od;
    [ 5, 60 ][ 6, 120 ][ 7, 210 ][ 8, 336 ][ 9, 504 ][ 10, 720 ][ 11, 990 ][ 12, 1320 ][ 13, 1716 ][ 14, 2184 ][ 15, 2730 ]

  [1]: https://i.sstatic.net/cNPii.png