So... is Btooom! finished or not?
NO, it is not finished.
Why?
The anime finished before the manga:
Anime finished: December 20, 2012
Manga finished: August 09, 2018
MangaUpdates contains information about the anime adaptation and indicates that it is incomplete.
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
Btooom! follows Ryouta Sakamoto, a NEET and avid gamer whose life is turned upside down when he wakes up on a mysterious island.
He soon discovers he's been unwillingly entered into a real-life version of his favorite online game, where the objective is to eliminate other participants using specialized bombs called "BIM."
Ryouta, initially opposed to violence, is forced to adapt and fight for survival. He eventually teams up with another player, Himiko, to navigate the dangers of the island and try to uncover the reasons behind this deadly game.
Together, they must use their skills and wits to survive against increasingly hostile opponents and find a way to escape the island and return home.