Kaiser and Klazar recently proved that all permutation classes of growth rate less than 2 are logarithmically asymptotic to the k-Fibonacci numbers for some k, and later Klazar proved that there are only countably many such classes.
I will discuss the permutation classes of growth rate a tiny bit larger than 2, precisely, those of growth rate less than the unique real root of 1 + 2x2 - x3, approximately 2.20557. I will prove (or, more accurately, indicate the proof) that there are only countably many such classes, although there are uncountably many classes of growth rate precisely approximately 2.20557. I might also give a complete list of their possible growth rates.