In inboard-limited plasmas, the scrape-off layer (SOL) shows two regions: the near SOL, extending a few mm from the last closed flux surface (LCFS), characterized by a steep gradient of the parallel heat flux radial profile, and a far SOL, typically some cm wide, with flatter heat flux profiles. The physics of the near SOL is investigated in TCV with two series of experiments featuring deuterium and helium plasmas, in which the plasma current, density and elongation have been varied. The parallel heat flux profiles are measured on the limiter by means of infrared thermography. For the first time, the near SOL is reported to disappear for low plasma current or at high density, for values of the SOL collisionality corresponding to a conduction-limited regime. The power in the near SOL is shown to decrease with the normalized Spitzer resistivity ν as . The floating potential profiles, measured at the limiter using flush-mounted Langmuir probes (LP), show the presence of non-ambipolar currents, and their relation to the presence of a velocity shear layer is discussed. The shearing rate is shown to strictly correlate with the power in the near SOL , consistently with a recent theoretical model. Measurements of the near SOL on the Low Field Side (LFS) are performed using a reciprocating Langmuir probe (RP). The near SOL is reported to vanish simultaneously at the LFS and at the limiter. The near and far SOL widths are compared with the predictions from existing theoretical models, to which empirical corrections with resistivity and elongation are proposed.