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Case with most HDD slots under $150
(alien.top)
We are digital librarians. Among us are represented the various reasons to keep data -- legal requirements, competitive requirements, uncertainty of permanence of cloud services, distaste for transmitting your data externally (e.g. government or corporate espionage), cultural and familial archivists, internet collapse preppers, and people who do it themselves so they're sure it's done right. Everyone has their reasons for curating the data they have decided to keep (either forever or For A Damn Long Time (tm) ). Along the way we have sought out like-minded individuals to exchange strategies, war stories, and cautionary tales of failures.
I really like the Fractal Design Node 804 which will hold 10 drives and will likely be the next case that I build in.
You could also go with the Phanteks Enthoo Pro 2 which should hold 12 drives.