This is just a stupid question about a good terminology. I'm interested in sequences $a_n$ with a growth that can be bounded by an arbitrarily small positive power of $n!$, i.e. for every $\epsilon > 0$ there should be a constant $c$ with $a_n \le c (n!)^{\epsilon}$. I wanted to call this growth "subfactorial" but then I learned that there is already a "subfactorial" $!n$. Any good suggestions?
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In the need of a name, I would also go for "subfactorial growth". In this case I see no danger of confusion; "sublinear" and "subexponential growth" are already used in analogous meaning (even if not completely standard). Also note that in your condition you can replace $n!$ with $n^n$ (which unfortunately didn't give me better hints) . 

