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Fornaess and Stout proved that EVERY complex manifold (connected and second countable) can be covered by finitely many relatively compact open subsets biholomorphic to a polydisc (Lemma II.1 in MR0470251). They even have an explicit bound on the size of the cover in terms of the dimension of the manifold. Further results of a similar flavour are contained in their papers MR0435441 and MR0662439.