When deposited from an evaporating solution onto a substrate, even nondescript nanoparticles can organize into intricate spatial patterns. Here we show that a simple but long-ranged anisotropy in nanoparticles' interactions can greatly enrich this scenario. In experiments with colloidal Co nanocrystals, which bear a substantial magnetic dipole, we observe assemblies quite distinct from those formed by nonmagnetic particles. Reflecting the strongly nonequilibrium nature of this process, nanocrystal aggregates also differ substantially from expected low-energy arrangements. Using coarse-grained computer simulations of dipolar nanoparticles, we have identified several dynamical mechanisms from which such unusual morphologies can arise. For particles with modest dipole moments, transient connections between growing domains frustrate phase separation into sparse and dense regions on the substrate. Characteristic length scales of the resulting cellular networks depend non-monotonically on the depth of quenches we use to mimic the effects of solvent evaporation. For particles with strong dipole moments, chain-like aggregates formed at early times serve as the agents of assembly at larger scales. Their effective interactions drive the formation of layered loop structures similar to those observed in experiments.