Are there molecules that take the shape of every platonic solid? Methane is the simplest molecule I know of that takes the shape of a platonic solid — in this case, the tetrahedron. Wikipedia has a page on platonic hydrocarbons that includes molecules with tetrahedron, cube, and dodecahedron shapes. There are lots of molecules that have octahedral geometry (e.g. sulfur hexafluoride). 

Are there any molecules that form an icosahedron? The closest I've found is $\ce{C_60}$, buckminsterfullerene, which takes the form of an Archimedean solid, the truncated icosahedron.

To be explicit: I am interested in molecules with atoms positioned at the vertices of platonic solids; whether or not there are direct bonds between those atoms is not relevant.
Yes, for each of the 5 platonic solids, there is at least one molecule that takes its shape.

There exist tetrahedron-shaped molecules. Some instances are $\text{CH}_4$ or $\text{CCl}_4$.

There exist octahedron-shaped molecules. Some instances are $\text{SF}_6$ or $\text{Mo(CO)}_6$.

There exist hexahedron-shaped molecules. An instance would be $\text{C}_8\text{H}_8$ AKA "cubane". Perhaps simple cubic lattices (like NaCl) could be considered "hexahedral molecules" as well?

There exist icosahedron-shaped molecules. An instance is elemental boron, $\text{B}_{12}$.

There exist dodecahedron-shaped molecules. An instance is dodecahedrane ($\text{C}_{20}\text{H}_{20}$).