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.