In 4d N=1, superpotential can only receive instanton corrections (hence only if there are vector multiplets).
In 4d N=2, Higgs branch has a hyperKahler metric, not renormalized, independent of Coulomb branch moduli; Coulomb branch is rigid special Kahler manifold independent of Higgs branch moduli. See http://arxiv.org/abs/hep-th/9603042 . The beta function is one-loop exact.
In 3d N=2, the superpotential depends on chirals but not on FI and real mass, the central charge depends on FI and real mass but not on chirals. See "Aspects of ..." paper.
In 3d N=3, R-symmetry is nonabelian, so in CFTs dimensions are not renormalized.
In 3d N=4, well-defined notion of Coulomb/Higgs branch, Higgs branch metric not renormalized.
In 2d N=(2,2), only divergence is logarithmic divergence of FI