In recent years the problem of describing regular covers of maps and abstract polytopes has attracted attention of several researchers. Of particular interest is the problem of determining for which abstract polytopes there exists a minimal regular cover, that is, a cover which is covered by any other regular cover of the polytope. It is known that if the polytope has rank 3 (polyhedron) then it has a minimal regular cover. Some examples with finite polytopes have been found. In this talk we present three examples with infinite starting polytopes, namely the Archimedean tessellation with triangular and hexagonal faces whose vertices have degree 4, the Archimedean tessellation with octagons and squares, and the one with triangles and dodecagons.