Background: Consider trying to cover the largest possible scaled copy of a planar region $C$ with specified shape with n instances of a tile $T$ of specified shape and size. Several families of this question - the largest disk that can be covered by n unit squares, largest square that can be covered with n unit disks and so on - are shown here: (https://erich-friedman.github.io/packing/index.html)
Question: In the best covering layout for a given $C$ with n $T$'s, consider regions where where more than one $T$ unit overlap. Observation: when tile $T$ is convex, no known best covering layout of any convex $C$ seems to have regions where more than 3 unit $T$s overlap. [Eg: see https://erich-friedman.github.io/packing/circovsqu/ - it shows covering largest possible squares with n unit circles: . For n = 3, 8 , 9, 11,... there are regions where three unit circles overlap but nowhere where more than 3 overlap]. Is 3 a provably strict bound for convex $T$s?
Note 1: When tile T is non convex, it appears that there can be regions in the best layout where arbitrarily large number of tiles overlap. An example is shown here: http://nandacumar.blogspot.com/2018/09/doubts-on-covering.html. This page also lists some related questions.
Note 2: And after the comments below, one can note that our question has a natural higher dimensional equivalent.
