Completism

Completism: A desire to obtain a complete collection of some type of thing.

Completist: A person who champions any philosophy which focuses on the completeness of something; as, a completist philosopher who holds that a definite complete body of knowledge exists to be discovered. A collector who strives to obtain a complete collection of some type of thing, especially a fan who aims to own or consume the complete output of a particular author, publisher, musician or band.