If I give you a natural number $n$, could you give me a topological manifold (of any dimension) that has exactly $n$ distinct differentiable smooth structures? For example, if $n=28$, one could give me $S^7$.
If so, what about restricting the dimension? Say, forcing all the manifolds to be dimension $257$ or something of the sort.
Here is what is within the realm of possibility for a fixed dimension:
(1) Producing examples of manifolds with a single smooth structure.
(2) Producing examples of manifolds with a finite, but nonzero amount of smooth structures.
(3) Producing examples of manifolds with infinitely many smooth structures.
(4) Given a fixed manifold $M$, computing the number of smooth structures on it.
Even restricting to spheres, this question is interesting and the subject of active research; see Groups of Homotopy Spheres: I, Detecting exotic spheres in low dimensions using coker J for the first two questions and any survey of the 4D Poincaré conjecture for the third. In theory, many of the techniques of 2 could be used for (4) for spheres, but the problem is a little harder than computing the stable homotopy groups of spheres, which is hard.