We consider the notion of cover complexity of finite languages on three different levels of abstraction. For arbitrary cover complexity measures, we give a characterisation of the situations in which they collapse to a bounded complexity measure. Moreover, we show for a restricted class of context-free grammars that its grammatical cover complexity measure w.r.t. a finite language L is unbounded and that the cover complexity of L can be computed from the exact complexities of a finite number of covers $$L' \supseteq L$$. We also investigate upper and lower bounds on the grammatical cover complexity of the language operations intersection, union, and concatenation on finite languages for several different types of context-free grammars.