Although Stampy's eye was considerably less swollen this morning (and the proud flesh had also gotten smaller) we took him to the vet, where he got Cephalexin for about 2 weeks (antibiotics) which will be an incredible 12 pills a day because he's so big, and one week of ointment on the gross part and in his eye 3x a day.
The vet said it could either be a puncture, a cyst, or an abscess; the cyst will come back but the other two won't. So we'll see.