For smooth manifolds it is known that they can admit a unique, finitely many, or a continuum of distinct smooth structures (I don't know whether there are any examples admitting precisely a countably infinite number).


For complex manifolds there are examples of smooth manifolds admitting a unique complex structure ($\mathbb{CP}^1$) or a continuum (compact Riemann surfaces, K3 surfaces, etc.) 



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> **Q.** Are there examples admitting only finitely many or a countably infinite number? 
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By deformation theory the tangent space to the moduli space of complex structures on $X$ should be given by $H^1(X, TX)$ (at least morally) so it must be necessary for this to vanish for every possible complex structure on $X$ to have any hope.



Examples of smooth manifolds admitting inbetween one and a continuum of complex structures
Examples of smooth manifolds admitting finitely many complex structures are provided by some holomorphically rigid complex varieties. 


For instance, considering [fake projective planes](https://en.wikipedia.org/wiki/Fake_projective_plane) (i.e., smooth compact complex surfaces which are not the complex projective plane but have the same Betti numbers as the
complex projective plane), there are $100$ examples determined up to biholomorphism, but only $50$ examples determined up to isometry. 


In fact, the real $4$-manifold underlying any fake projective plane admits precisely *two* inequivalent complex structures, that are exchanged by complex conjugation.


Moreover, by standard results on rigidity, any minimal Kähler surface with the same fundamental group as a fake projective plane is actually biholomorphic or conjugate biholomorphic to it. 


