The glass transition in binary mixtures of star polymers is studied by mode coupling theory and extensive Molecular dynamics computer simulations. In particular, we have explored vitrification ill the parameter space of size asymmetry delta and concentration rho(2) of the small star polymers at fixed concentration of the large ones. Depending on the choice of parameters, three different glassy states are identified: a single glass of big polymers at low delta and low rho(2), a double glass at high delta and low rho(2), and a novel double glass at high rho(2) and high delta which is characterized by a strong localization of the small particles. At low delta and high rho(2) there is a competition between vitrification and phase separation. Centered in the (delta, rho(2))-plane, a liquid lake shows up revealing reentrant glass formation. We compare the behavior of the dynamical density correlators with the predictions of the theory and find remarkable agreement between the two.