Rhombicosidodecahedron

A dodecahedron has $20$ vertices, $30$ edges and $12$ faces.
A isocahedron has $12$ vertices, $30$ edges, and $20$ faces.
Hence, a rhombicosidodecahedron, combining both and adding some square faces (see helpful illustration provided above), has

$\dfrac { 1 }{ 2 } \left( 3\cdot 20+5\cdot 12 \right) =60$ Vertices
$2\left( 30+30 \right) =120$ Edges
$12+20+\dfrac { 1 }{ 2 } (60)=62$ Faces

Hence

$V-E+F=60-120+62=2$

Of course we can use the general result of $2$ of the Euler Characteristic for any convex polyhedron, but what's the fun in that?

Euler's characteristic states that, on any convex polyhedron, $V-E+F$ is always $\boxed{2}$ .