Closing the loop around an exponentially stable single-input/single-output regular linear system, subject to a globally Lipschitz and nondecreasing actuator nonlinearity and compensated by an integral controller, is known to ensure asymptotic tracking of constant reference signals, provided that: 1) the steady-state gain of the linear part of the plant is positive; 2) the positive integrator gain is sufficiently small; and 3) the reference value is feasible in a very natural sense. Here lower bounds are derived for the maximal regulating gain for various special cases including systems with nonovershooting step-response and second-order systems with a time-delay in the input or output. The lower bounds are given in terms of open-loop frequency/step response data and the Lipschitz constant of the nonlinearity, and are hence readily obtainable.