completist

Short definition from Wiktionary

Noun

1. 1
   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.
2. 2
   A person who champions any philosophy which focuses on the completeness of something.

Adjective

1. 1
   Which attempts to collect something completely.
2. 2
   Alternative spelling of completest.