The asymmetric inclusion process is a lattice-gas model which replaces the "fermionic" exclusion interactions of the asymmetric exclusion process by "bosonic" inclusion interactions. Combining together probabilistic and Monte Carlo analyses, we showcase the model's rich statistical complexity-which ranges from "mild" to "wild" displays of randomness: gaussian load and draining, Rayleigh outflow with linear aging, inverse-gaussian coalescence, intrinsic power-law scalings and power-law fluctuations and condensation.