Ring compounds of non-carbon atoms
Pentazole exists - an $\ce{N5H}$ ring. It is stabilised by aromaticity, with 6 $\pi$-electrons in a cyclic, planar system.



Sulphur forms many allotropes which are rings but these do not contain hydrogen. $\ce{S8}$ and $\ce{S7}$ are the most common.

Phosphorus also forms cyclic allotropes such as white phosphorous, $\ce{P4}$.



DavePhD also mentioned in his answer that a silicon analogue of benzene has been synthesised recently but it is not $\ce{Si6H6}$ because it has other groups attached to some of the silicon atoms.