The act of asking your partner to ram their head up your anus during intercourse or as a type of foreplay, as opposed to ramming their fist up your anus. The term comes from observing how goats tend to ram their head into other people's butts in cartoons and the like.
John:"So... Sarah said:"goat me" while we were having sex yesterday. Isn't that crazy"
Peter:"Dude, that's so totally sick. I wish my girlfriend would ask me that. You are so lucky!"