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Is there a notion of a context-free complete language (in the analogous sense to a $NP$-complete language)?

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Yes.

Lautemann and Schwentick prove that Greibach's "hardest context-free grammar" with a neutral symbol is complete for $LOGCFL$ and hence $CFL$ also, under quantifier-free projection without BIT.

This is Corollary 4.3 in their paper

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