  We classify these threefolds, which are the ones such that their universal
cover is not compact and not covered by positive-dimensional compact analytic
subsets. We show that these threefolds have nonnegative Kodaira dimension, and
that their Iitaka-Moishezon fibrations are, after a suitable finite etale
cover, bimeromorphic to a submersion with fibres complex tori, and base of
general type and $\pi_1$-general type.
