Although it is tempting to assume that $\psi^*(x)\psi(x)$ must manifest some substantive property beyond that of a mere calculating device--as advocated by the Copenhagen Interpretation, there is a simple way of rebutting its most obvious reification.
Assuming that, $\psi^*(x)\psi(x)$, is an actual density consisting of a "spread out particle", then for an electron in a hydrogen ground state, one might reasonably expect that the quantity $-e\psi^*(x)\psi(x)$ would play the role of a physically meaningful charge density. That is, such a charge density should be the source of an electric field. Moreover that field would be associated with a total energy equal to the repulsive self-interaction potential energy arising from the integral over all pairs of infinitesimal sub-volumes.
In other words, it would possess a capacitor like self-energy about equal to 1/2 e squared divided by a Bohr radius which plays the role of the effective capacitance. Since this energy would nearly cancel the negative binding energy, it would significantly decrease the Hydrogen ionization energy to the point of making the H atom (and the rest of the chemical elements) unstable --in contradiction of experiment. This is the reason why the Hartree approximation only includes inter-actions between different electron charge densities.
The main question, then, is how might it be possible to retain an effective charge density interpretation of $-e\psi^*(x)\psi(x)$ in light of the problem above.