The capability of predicting the density limit of a magnetically confined burning plasma is of crucial importance to establish the ultimate performance of a fusion power plant. The Greenwald density limit, commonly used as an empirical scaling law, predicts that the maximum achievable central line-averaged density is given by the relation , where is the average plasma current density and k is the plasma elongation. However, several experiments have pointed out that such a limit can be overcome in the presence of peaked density profiles. This paper proposes a new empirical scaling law for a limiter tokamak operating in the low-energy confinement mode (L-mode) concerning the case of peaked density profiles associated with the presence of multifaceted asymmetric radiation from the edges. This result is based on dedicated experiments performed on the Frascati Tokamak Upgrade (FTU) under extremely clean machine conditions (Zeff = 1.0–1.5), in which the high-density domain is explored in a wide range of values of plasma current (Ip = 500–900 kA) and toroidal magnetic field (BT = 4–8 T). It is found that the maximum achievable central line-averaged density essentially depends on the toroidal magnetic field only and does not depend on the average plasma current density: the behaviour is explained in terms of density profile peaking in the high-density domain. As a confirmation that the limit is an edge limit, it is also shown that a Greenwald-like scaling (i.e. depending on the current density) actually holds for the edge line-averaged density (at r/a ≃ 4/5).