We investigate how varying molecular topology of polymers influences crystallization in thin polymer films. In particular, we simulate linear and star polymers of fixed mass having a progressively increasing number of arms (f ≤ 16) in a system where the linear polymer exhibits crystallization in a thin film geometry, but no apparent crystallization in the corresponding bulk material. The degree of crystallization of the polymer film at long times decreases progressively with increasing f, and no crystallization is observed beyond f = 8. Crystallization for smaller values of f develops as a sigmoidally shaped wavefront initiating from the supporting crystalline interface. We suggest that large shape fluctuations and the competition of length scales of star polymers with high f lead to inhibited crystallization.