Significant progress has been made in the last decade in understanding mixtures of hard–sphere colloids and (smaller) non–adsorbing ideal, linear polymers. We introduce extra complexity into this simple model system by replacing the linear polymers wit star–branched polymers with increasing functionality but constant radius of gyration. The observed phase diagrams, interprete in light of what is known about hard–sphere colloid plus linear polymer and binary–hard–sphere mixtures, suggest that 32–ar stars are close to behaving hard–sphere–like in colloid–star mixtures at this size ratio.