2D Ruddlesden-Popper-phase perovskites (2D RPPs) have recently been shown to not only exhibit excellent photoabsorptive properties for light harvesting and photovoltaics, but also a strong third-order nonlinear response. This work is about a more detailed characterisation of this third-order nonlinearity of 2D RPPs with different chemical composition around their excitonic bandgaps using a Z-scan technique. As the key result the 2D RPPs are found to exhibit a strong enhancement of the nonlinear response and a sign change of the nonlinear refractive index around the excitonic bandgap with peak values on the order of 10^-14 m^2/W for n2 and -0.1 cm/kW for beta_eff, which is several orders of magnitude higher than previously reported values for highly nonlinear chalcogenide glasses. The investigated flakes differ in their inorganic layer thicknesses and hence exciton confinement, resulting in different spectral positions of the excitonic bandgap. In terms of their chemical composition, the flakes are lead halide RPPs, with both iodide and bromide-containing flakes being studied. The spectral positions of the excitonic bandgap in these flakes cover most of the visible spectrum, with bandgaps ranging from about 400 nm to 650 nm. The optical response at the excitonic bandgap is dominated by saturable absorption for all flakes. This effect was particularly prominent in a lead halide flake with the inorganic layer just consisting of one monolayer, with a more than 15x increase in transmission. In summary, the studied 2D RPPs exhibit an extremely strong nonlinear optical response in the visible and are promising candidates for applications in all-optical switching or frequency conversion.