Three-body nonadditive forces in systems of three spin-polarized alkali-metal atoms (Li, Na, K, Rb, and Cs) are investigated using high-level ab initio calculations. The nonadditive forces are found to be large, especially near the equilateral equilibrium geometries. For Li, they increase the three-atom potential well depth by a factor of 4 and reduce the equilibrium interatomic distance by 1.07 Angstrom. The nonadditive forces originate principally from chemical bonding arising from sp mixing effects.