Can't you form a ring from almost any number of sides from carbon?
Carbon bonds have certain preferred angles. For example, the tetrahedrally symmetrical methane, which has sp3 hybridization,, $\ce{CH4}$, has an angle of 109.5 0. In benzene, where there is sp2 hybridization, the angle is 120 0.

These are the preferred angles for bonds. In some molecules, the geometry forces them to odd angles, such as cyclopropene, with a 60 0 bend. This imposes stress, much like making Tinker toy models where you have to bend the sticks a bit. Overdo this and the stick (bond) snaps.

So, here are some cyclic (ring) compounds with the following number of carbons, and at least one alkene bond:

3: cyclopropene

4: cyclobutene

5: cyclopentene

6: cyclohexene

7: cycloheptene


cyclooctene


Notice in the drawings of the molecular structure that larger rings are increasingly asymmetrical, degenerating to a twisted mess. So yes, very large carbon "rings" are possible, but eventually cannot hold their shapes. Some compounds contain more than on C=C bond, such as 1,3-cyclohexadiene, but I know of no cyclic compounds where all carbons contain a double (alkene) bond. 

You can extend this, though, to build up three-dimensional polyhedra, such as cubene or the elegant (though not double-bonded) buckminsterfullerene.