A commenter below identified the mystery weed as lichweed, or Parietaria officinalis. Brambles are evil - there was a big patch in our field that we mowed to the ground and then built a giant bonfire on top of the remains. They came back. We put the goat pen right over the patch, and let the goats get very bored - when goats are bored, they eat. Somehow, the blackberries are still there.