The existing group formation-containment (GFC) studies for multi-agent systems (MASs) depend on system model information and generally neglect input saturation constraint, thereby limiting their applicability to MASs with unknown system model and input saturation. This paper investigates the data-driven GFC control problem of nonlinear MASs with asymmetric input saturation. First, a novel communication topology selection algorithm with relaxed topology conditions is proposed. Then, to tackle the challenge posed by asymmetric input saturation, a novel nonquadratic performance index function with a simplified formulation is designed and the corresponding Hamilton-Jacobi-Bellman equation is derived. On this basis, an effective value iteration algorithm is proposed to determine the optimal GFC control policy, accompanied by the rigorous mathematical analysis. By establishing the critic-actor framework based on generalized fuzzy hyperbolic model, a novel data-driven algorithm is proposed to achieve GFC under asymmetric input saturation, which overcomes the dependence on system model. Finally, some simulation results are provided to verify the effectiveness and superiority of the proposed data-driven GFC algorithm.