Unlike most answers I've found, this valve is not leaking at the seat washer. I still replaced that since I had the valve open. I tried putting some packing inside the concentric nuts, the silver one that holds the cover and the bronze one that the silver one goes into. Still I cannot stop the drip which only happens when the valve is opened to run the water.
The valve while leaking:
The only discernible markings are around the stem with the word WOLVERINE stamped into the metal.
Here is a picture of the valve stem for further reference:
And here is a picture of the manifold the valve goes into behind the wall:
Best Answer
I did as Jack mentioned in the comments. I got some graphite packing compound and placed it around the stem between the two brass nuts. After tightening all down, some of the packing squeezed out around the stem and the silver nut where the drip was. That has stopped the water for now. I guess this is going to be an ongoing maintenance task until the packing wears away.