Hadith No. 1568 (below) is a repeat. Read it here.
Narrated Ya'li (radiallaahu `anhu):
While I was with Allah's Messenger () there came to him a man wearing a cloak having a trace of yellowish perfume or a similar thing on it. `Umar used to say to me, "Would you like to see the Prophet () at the time when he is inspired divinely?" So, it happened that he was inspired (then) and when the inspiration was over the Prophet () said (to that man), "Do in your `Umra the same as you do in your Hajj."
Narrated Ibn Abbas (radiallaahu `anhu):
A man bit the hand of another man but in that process the latter broke one incisor tooth of the former, and the Prophet () forgave the latter.
Two people were fighting, one bit the hand of the other, while the latter broke the tooth of the former while snatching away his hand. When they went to the Prophet (sallallaahu `alayhi wasallam) to judge between them, he forgave the latter (who had snatched away his hand) and said that there will be no diyat (blood money: 5 camels for one tooth) in this case. He said:
يَعَضُّ أَحَدُكُمْ أَخَاهُ كَمَا يَعَضُّ الْفَحْلُ، لَا دِيَةَ لَك
[roughly translated:] "One of you bit the hand of the other like a horse/stallion bites. There's no diyat (blood money) for you." [Bukhari & Muslim]
So the scholars have said that there's no kaffaarah for the one who bit the hand. Also, the one who snatched away his hand should've freed himself some other way instead of breaking the other's tooth.
P.S. I don't know why, but this incident seems sort of funny to me.