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The name Kenpachi is inherited by only the strongest Soul Reaper. Tradition has the current Kenpachi leads Eleventh Squad of the Thirteen Court Guard Squad. Tradition also demands that to claim the name Kenpachi, a Soul Reaper must challenge the current Captain to the death in front of a thousand Unseated Soul Reapers.
The First Kenpachi, Yachiru Unohana- later changed her name to Retsu- was a member of the original Thirteen. They weren't the Soul Reapers of today, they were killers. Monsters. All they did was kill and destroy anything that stood in their way.
After batting a nameless boy from the Zaraki district, she decided to surrender the name Kenpachi- because she wanted to battle him again and she only wanted to battle him again. She didn't want to waste her time on the worthless enemies she so effortlessly sliced apart. So she went to the Forth Squad, and became a Medic. She gave her name up to her former Lieutenant.
The Second Kenpachi, Atsuji Makihara. Former Lieutenant of Unohana. Diligent Captain. Was killed by who would later become the Third Kenpachi.
Third Kenpachi, Yuki Kashin. He was open to learn, but he had no skill that a Captain needed. He dropped out after a few months. He let himself be defeated- thinking he was no longer suited to be named Kenpachi.
Fourth Kenpachi, Fujimoto Himura.
Fifth Kenpachi, Shigure Osaka. She was a terrorist. She was put to death by the Soul Society after she tried to assassinate the Head Captain.
Sixth Kenpachi, Tohru Takayi. She was Shigure's Lieutenant. After she was interrogated to make sure she wasn't working with Osaka. When we deduced that she wasn't, she was given the name Kenpachi.
Seventh Kenpachi, Yamai Kobashi. Nothing much to mention about him. He was skilled with a sword, and he was a decent Captain as well.
Eighth Kenpachi, Junichi Sendo. A strange case. He grew up in the Rukon District, and he managed to battle through them without the aid of a Zanpakuto or any weapon. He became Captain with just his fists. He killed more challengers than any Kenpachi. He was challenged so much because majority of people didn't think a Soul Reaper without a Zanpakuto was fit to be the Captain.
Ninth Kenpachi, Kyo Naniwa. One of the few Kenpachi known to have an elemental Zanpakuto. He used a Fire Zanpakuto. He defeated Sendo by burning him alive before he could even raise his fists. A cold-blooded killer.
Tenth, Gosuke Kiganjo. A lazy bastard. Did no work with his squad. Only wanted to fight and kill. He failed to follow any of his other Captain's duties. He was very unpopular with the other Captains and the rest of his Squad. Including his Lieutenant, who tried to poison him on multiple occasions.
Eleventh, Kenpachi Zaraki. He was the boy Unohana battled. He had no name. He grew up in the Zaraki district of Rukon. The worest district. He had to kill people to become stronger- in order to survive.
A few hundred years ago, he dueled Unohana in a battle to the death. She was killed, obviously.

"Yeah?" Asked a bored voice. "Why are you telling me this, old man?"

Shunsui Kyoraku, the current Head Captain, looked at him. "Zaraki is now an old man. Someone needs to take his place soon. And I want that to be you, Genji Araki."

Genji grinned.

"Will you battle Zaraki?"

"Hell yes!"