Construction of a Via
Mister A-Dog, you need to look up a diagram and think about how a via is constructed, and what it's true shape is. The annular ring is a circular pad on one layer of a PCB. It becomes a ring when a drill hole is made through the center. You need a ring on top and bottom layers, and any other layers where a trace connects to the via. After these rings in the padstack have been drilled, they are connected by a copper plating process that forms a conductive tube on the inner wall of the drill hole. Hence Willbe's comment on wall thickness being the current limiting feature. Wall thickness around 1 mil is pretty standard. The size of the ring is limited mainly by how accurately the drill can be placed in the center of the pad. The ring will typically be at least 5 mils wide without getting into premium fabrication methods. To some extent the ring should be wider for larger vias.
So, in general, it is the size of the plated hole that determines how much current can be carried. Too much current will heat the via. A safe rule of thumb for a rough estimate is 1 amp of current per 10 mils diameter of via. Beyond two or three amps, use multiple vias.
Hope this answers your question!
-Brock