  The capability to isolate one to few unit-cell thin layers from the bulk
matrix of layered compounds opens fascinating prospects to engineer novel
electronic phases. However, a comprehensive study of the thickness dependence
and of potential extrinsic effects are paramount to harness the electronic
properties of such atomic foils. One striking example is the charge density
wave (CDW) transition temperature in layered dichalcogenides whose thickness
dependence remains unclear in the ultrathin limit. Here we present a detailed
study of the thickness and temperature dependences of the CDW in VSe$_2$ by
scanning tunnelling microscopy (STM). We show that mapping the real-space CDW
periodicity over a broad thickness range unique to STM provides essential
insight. We introduce a robust derivation of the local order parameter and
transition temperature based on the real space charge modulation amplitude.
Both quantities exhibit a striking non-monotonic thickness dependence that we
explain in terms of a 3D to 2D dimensional crossover in the FS topology. This
finding highlights thickness as a true tuning parameter of the electronic
ground state and reconciles seemingly contradicting thickness dependencies
determined in independent transport studies.
