The Serre closure of Banach spaces

We determine the Serre closure of Banach spaces in the category of quasi-Banach spaces, that is, the smallest class of quasi-Banach spaces that contains all Banach spaces and is stable under subspaces, quotients, and extensions. Precisely, a quasi-Banach space $X$ belongs to this closure if and only if there exist constants $M$ and $k\in\mathbb N$ such that $$ \frac{\|x_1+\dots+x_n\|}{\|x_1\|+\dots+\|x_n\|}\leq M\log^k n $$ for every $n\geq 2$ and every $x_1,\dots,x_n\in X$; equivalently, $X$ is a quotient of an Orlicz sequence space $\ell(I,f_l)$, where $I$ is an index set and $f_l(t)$ is equivalent to $t\log^l(1/t)$ near zero. The proof combines geometrical estimates for quasi-Banach spaces and quasilinear maps with the study of higher order derived spaces associated to complex interpolation between $\ell_{2/3}$ and $\ell_2$. We provide examples of spaces in the Serre closure of Banach spaces that are not iterated extensions of Banach spaces and discuss some related problems.