Silver is the most electrically conductive element on the Periodic Table. However, I am interested in knowing if there are any compounds (combinations of elements) that would be more conductive than Silver (without Superconductor cooling). The extreme electrical conductivity of specific chemicals has many real world applications, from simply calling someone miles away, to processing enormous amounts of data in servers.
Graphene supposedly has higher conductance.
The [...] resistivity of graphene sheets would be 10−6 Ω⋅cm. This is less than the resistivity of silver, the lowest otherwise known at room temperature.
but it is not a compound - it is a homonuclear molecule.
The short answer is "no". There are materials, both actual and theoretical, which would have a higher conductivity, but not as a simple "material". They only make better conductors under very precise conditions. Since you specifically ruled out superconductive cooling I think we can also safely rule out materials that are a single atom thick as a prerequisite for great conductivity.
Graphene, for example, is known as a two dimensional or "2D" material. Its conductive properties only materialize when it is very, very thin. A single atom thick. You're not going to go out and buy a spool of graphene wire and expect it to be the best conductor ever.
Stanene is a relatively new discovery. So new, in fact, it hasn't even been produced or tested yet, though experiments are currently underway. It, also, is a 2D material made from a single layer of tin atoms. Tin is a "topological insulator", meaning that its interior structure isn't a great conductor, but its surface is a very good conductor. Electrons all flow on the outer surface. In electronics dealing with alternating current this is know as the "Skin Effect". So, theoretically, if it were a single atom thick it would be all surface and no interior, making it a better conductor than even graphene. Researchers are actually claiming 100% efficiency, a very bold claim.
So no, there are no "materials" which are better conductors, at least that I can find. There are, however, "atomic structures" which are better conductors. But since you're asking this question I assume you are not a microchip manufacturer, making these materials pretty much useless to you, personally. If you are a microchip manufacturer then, yeah, there's some exciting stuff out there. Otherwise, though, there is still nothing better than silver.