We provide evidence that gallium purity is the primary impediment to attainment of ultra-high mobility in a two-dimensional electron gas (2DEG) in AlGaAs/GaAs heterostructures grown by molecular beam epitaxy (MBE). The purity of gallium can be enhanced dramatically by in-situ high temperature outgassing within an operating MBE. Based on analysis of data from an initial growth campaign in a new MBE system and modifications employed for a 2nd growth campaign, we have produced 2DEGs with low temperature mobility μ in excess of 35×106cm2/Vs at density n=3.0×1011/cm2 and μ=18×106cm2/Vs at n=1.1×1011/cm2. Our 2nd campaign data indicate that gallium purity remains the factor currently limiting μ<40×106cm2/Vs. We describe strategies to overcome this limitation.