So... is Beck finished or not?
NO, it is not finished.
Why?
The anime finished before the manga:
Anime finished: March 31, 2005
Manga finished: June 05, 2008
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
Beck is a coming-of-age story centered on Koyuki, a bored and aimless teenager whose life takes an unexpected turn when he rescues a dog and meets Ryuusuke, a talented guitarist with big dreams.
Inspired by Ryuusuke's passion and the allure of rock music, Koyuki embarks on a musical journey of his own, learning to play the guitar and joining Ryuusuke's new band, BECK.
The anime follows BECK's formation and their efforts to make a name for themselves in the Japanese music scene. They face challenges and celebrate triumphs as they strive to achieve their rock and roll ambitions.
Essentially, it's about the power of music to transform lives and the dedication needed to pursue a dream, all while navigating the complexities of friendship and growing up.