Are there noncompact complex manifolds that a) Don't embed in C^n (holomorphically) and b) Cannot be covered by a finite number of coordinate open sets? If b) can be satisfied, then I think so can a) be by taking a product with a compact complex manifold. If one takes a Riemann surface of infinite genus, one does not have a "good" finite open cover, but I allow noncontractible open covers as well. Apologies in advance for this elementary question.

Fornaess and Stout proved that EVERY complex manifold (connected and second countable) can be covered by finitely many 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. 


If $\widetilde X$ is a compact complex manifold of dimension $\geq 2$ and $x \in \widetilde X$ then $X = \widetilde X  \lbrace x \rbrace$ is a noncompact manifold that cannot be holomorphically embedded in $\mathbb C^N$. This is because, by Hartogs' Theorem, we have $\mathcal O(X) = \mathcal O(\widetilde X)$ and therefore global holomorphic functions on $X$ are constant, which is not the case for complex submanifolds of $\mathbb C^N$. 


Part a) is evident for the blownup manifold $X$ since it contains onedimensional compact submanifolds. As for part b), I am guessing (but cannot prove) that the huge first Chern class of the line bundle $\mathcal O_X(D)$ associated to the exceptional divisor $D$ prevents the existence of a finite number of holomorphic or even differentiable charts. 


It depends probably a bit on the notion of "chart" but if you allow your charts to have countably infinitely many connected components then it is a consequence of dimension theory that a $n$ dimensional manifold can always be covered by $n+1$ (?) charts even in such a way that the connected components of arbitrary intersections are either empty or contractible. This is not quite a "good" cover, but comes very close. In particular, vector bundles trivialize locally over such a finite atlas. You can find this in e.g. Well's book of complex manifolds. I don't have a copy here, so I can't tell you the precise page but you surely will find it. 


I suppose most (which?) complex $n$manifolds can be classified by a degree $p\in\mathbb{N}$, that of holomorphic completeness. The simplest way I see it is as the dimension $p1$ of the compact factor on a product of a compact complex manifold by $\mathbb{C}^{np+1}$. I see it also as the maximal dimension (+1) a compact complex submanifold can attain... Stein manifolds=holomorphic 1complete=holomor. embeddable in some $\mathbb{C}^n$. Andreotti, Cartan, Grauert, Remmert, Stein worked a lot on the notion in the middle of XXth century. A consequence is that every coherent sheaf's $i$cohomology vanishes for $i\geq p$. But how could we deduce the number of charts from cohomology? For which sheaf? 

