We examined 11 difficult reactions with self-interaction corrected density (SIC) functional theory. The data set includes dissociation of radicals into symmetric fragments (H-2(+)-->H+H+, He-2(+)-->He+He+), radical hydrogen abstraction (H+H-2-->H-2+H, H+HCl-->H-2+Cl, H+N2H2-->N2H+H-2, CH3+H-2-->CH4+H), proton transfer [HC(OH)CHC(O)H-->HC(O)CHC(OH)H], S(N)2 halogen exchange (X-+CH3X-->CH3X+X-, X=F,Cl,Br), and closed-shell unimolecular dissociation of tetrasine (C2N4H2-->N-2+2HCN). Calculated self-interaction energies cancel, almost identically, for the reaction energies (DeltaE(R)), so that SIC functionals do not lead to a systematic improvement in DeltaE(R). Self-interaction correction increases for reaction transition structures, leading to higher calculated activation barriers (DeltaE(not equal)). The average absolute deviation in DeltaE(not equal), from ab initio and experimental barriers, is reduced from 14 kcal/mol for Vosko-Wilk-Nusair (VWN) or 12 kcal/mol for revised Perdew-Burke-Ernzerhof (revPBE) functionals to 5.4 (SIC-VWN) or 3.4 (SIC-revPBE) kcal/mol. Reorganization of the electron density, due to removal of self-interaction, appears to be important. When SIC is included as a perturbation, using self-consistent densities of the parent functional, the average absolute deviations for the barriers increase to 7.5 (VWN+SIC) or 5.3 (revPBE+SIC) kcal/mol. Gradient-corrected functionals (revPBE, BP86) reduce the magnitude of the total self-interaction correction, by improving the description of the core orbitals. For the valence orbitals, both the magnitudes of the self-interaction corrections, and their change between reagents and transition structures, are similar for VWN local density approximation, and generalized gradient approximation functionals. Reducing the magnitude of the self-interaction energy for valence electrons thus appears to be a promising direction for the development of chemically accurate exchange-correlation functionals.