You can find a polyhedron with any number greater than 4 of vertices or faces.
However, a torus, ellipsoid, sphere, paraboloid, hyperboloid are all standard shapes with no vertices. Cylinders, too, have no vertices. And there are many completely random shapes - a lump of putty, for example, which will have no vertex.
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A cylinder and a cone - are two entirely different 3D shapes. A cylinder has three faces & two edges. A cone has two faces and one edge.
There are infinitely many shapes that do.
The rule applies to POLYHEDRA (3D shapes) not Polygons, which are 2D Faces + Vertices - Edges = 2
A solid cone has 2 faces and 1 edge
A pyramid with an n-sided base will have n + 1 vertices, n + 1 faces, and 2n edges.