jmc

The original prism has 5 faces, 9 edges, and 6 vertices. If the new pyramid is added to a triangular face, it will cover one of these faces while adding 1 new vertex, 3 new edges, and 3 new faces. If instead the new pyramid is added to a quadrilateral face, it will cover one of these faces while adding 1 new vertex, 4 new edges, and 4 new faces. So, we maximize the sum by adding a pyramid to a quadrilateral face. This gives us a solid with $5-1+4=8$ faces, $9+4=13$ edges, and $6+1=7$ vertices. The sum of these is $\boxed{28}$.