So... is Busou Renkin finished or not?
YES, it is finished.
Why?
The anime finished after the manga:
Anime finished: March 29, 2007
Manga finished: April 25, 2006
MangaUpdates contains information about the anime adaptation and indicates that it is complete.
Watching Order (by release date)
Here you have all the anime relations we detected ordered by release date, so you can see the reason for this status in context and also use it as watching order.
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Synopsis
Busou Renkin tells the story of Kazuki Mutou, an ordinary student who stumbles upon a battle between a girl named Tokiko Tsumura and a monster called a homunculus.
During the encounter, Kazuki unknowingly receives a powerful weapon called a Kurogane.
This weapon, manifested as a Busou Renkin, grants him special abilities.
Now, Kazuki is thrust into a hidden world where alchemically created monsters prey on humans.
He must learn to control his newfound powers and fight alongside Tokiko to protect humanity from the homunculi.
His life is forever changed as he struggles to balance his old life with his new responsibility as a warrior.