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A chess opening(s) database contains the series of moves that make up the different openings.
A database could hold many different openings. It is one database.

Without the "chess" part, what should it be called,
opening database
or
openings database
or
openings' database.

Thanks.
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Openings gets my vote.
Comments  
That's the one I would choose but I'm being outnumbered by my chess friends who reckon it should be "opening" database.