A theory of the non-diffusive anisotropic kinetic alpha-effect (“Γ-effect”) for densitystratified rotating turbulent fluids is developed. No limitations on the rotation rate are imposed and the fully nonlinear dependence of the Γ-effect on the angular velocity is studied. When the Coriolis number, ω* = 2τ ω, is small the dimensionless “dynamo number”, Cτ, characterising the power of the Γ-effect, grows with ω*. The dependence, however, reaches a maximum for ω* ∼ 2. For still higher rotation rates CΛdecreases as 1/ω*. In opposition, the corresponding number, Cx, of the hydromagnetic α2-dynamo problems remains finite for very large ω*. Hence, for fast rotation the hydrodynamic Γ-effect is small while the hydromagnetic α-effect remains large. In consequence, the large-scale magnetic and velocity structures are expected to be generated with roughly equal power in slowly rotating objects. In the rapid rotators, however, generation of the large-scale flows is problematic.