The commonly adopted scaling for the maximum achievable plasma density in tokamak fusion devices, the so-called 'Greenwald limit', refers to the line-averaged density along a central chord and depends only on the average plasma current density. However, the Greenwald limit has been exceeded in tokamak experiments in the case of peaked density profiles, indicating that the edge density is the real parameter responsible for the density limit. Furthermore, the Greenwald limit has been obtained for fixed density profiles, so that the scaling can be very different when introducing density profile dependencies on plasma parameters. Dedicated density limit experiments were performed in recent years on the Frascati Tokamak Upgrade, exploring the high density domain in a wide range of values of plasma current, toroidal magnetic field and edge safety factor. New data were collected in the latest experimental campaigns, extending the study of the density limit towards lower values of toroidal magnetic field and plasma current. These experiments confirmed the edge nature of the density limit, as a Greenwald-like scaling was obtained for the maximum achievable line-averaged density along a peripheral chord, while a clear scaling of the maximum achievable line-averaged density along a central chord with the toroidal magnetic field only was found and successfully interpreted as due to interplay between the peripheral Greenwald limit and the specific density profile behavior when approaching the density limit. In particular, an analytical relation between the peripheral and the central density limit was derived for the first time, with the introduction of a generalized parabolic density profile with the peaking factor dependent on the plasma parameters.