We report results on the transient and the instantaneous nonlinear optical (NLO) response of C₆₀ and the higher fullerenes C₇₀, C₇₆, C₈₄ dissolved in toluene using, respectively, the z-scan technique and energy-dependent transmission measurements with 532 nm, 10 ns laser pulses, and the optical Kerr effect with 800 nm, 100 fs laser pulses. The NLO measurements on C₇₆ are the first reported for this fullerene. The second hyperpolarizability γ of the fullerenes has a negative sign in both nanosecond and femtosecond pulse regimes. The transient response value of γ(C₇₀) and the instantaneous response of γ(C₇₆) were much larger than expected from the theoretically predicted smooth dependence of γ on fullerene size. The degree of validity is discussed for factors such as molecular symmetry, molecular volume, aromaticity and other molecular properties previously considered as determining fullerene-size dependence of γ. The observed trends and anomalies are explained, for the transient response, in terms of electronic-state population-dependent photophysical properties and, for the instantaneous response, by inclusion of photon resonance effects.