The internal angles at any given vertex have to be less than 360 degrees (or else the shape is flat).
Also at any given vertex, 3 or more faces are meeting.
So the only combinations of 3+ regular polygons that have additive internal angles of less than 360 degrees are:
3 triangles (60x3), aka, Tetrahedron
4 triangles (60x4), aka, Octahedron
5 triangles (60x5), aka, Icosahedron
3 Squares (90x3), aka, Cube
and 3 Pentagons (108x3), aka, Dodecahedron
You can't use shapes bigger than Pentagons because 3 hexagons would be 120x3=360=flat