  We study almost complex structures with lower bounds on the rank of the
Nijenhuis tensor. Namely, we show that they satisfy an $h$-principle. As a
consequence, all parallelizable manifolds and all manifolds of dimension
$2n\geq 10$ (respectively $\geq 6$) admit a almost complex structure whose
Nijenhuis tensor has maximal rank everywhere (resp. is nowhere trivial). For
closed $4$-manifolds, the existence of such structures is characterized in
terms of topological invariants. Moreover, we show that the Dolbeault
cohomology of non-integrable almost complex structures is often infinite
dimensional (even on compact manifolds).
