Let $M$ be the fake $CP^2$ (namely the closed topological 4-manifold which is homotopy equivalent but not homeomorphic to the complex projective plan). It is well-known that $M$ admits no smooth structures. However $M' = M - pt$ is smoothable by a theorem of Quinn. The question is: does someone known an explicit handlebody decomposition of $M'$?
There are not that many explicit handlebody pictures of exotic open 4-manifolds, because they get awfully complex in short order. The ones that I know of are in work of Žarko Bižaca from the mid-90's. I think you probably can work out this particular case by hand. You don't really want to use Quinn's theorem for this, because it is not exactly constructive. On the other hand, Freedman's construction produces this manifold in a somewhat explicit manner.
Here is a sketch. Start with a 4-ball, and then attach a (+1) framed 2-handle along a trefoil knot. If you've chosen the correct trefoil, the boundary of the resulting manifold is the Poincare homology sphere, say P. Then Freedman tells you that P is the boundary of a (topological) contractible manifold W, which you glue on to make the exotic $CP^2$, commonly known as Ch (for Chern). The reason that you don't need Quinn is that the construction of W is done by making a manifold $W'$ that is proper homotopy equivalent to $S^3 \times [0,\infty)$, and then using the proper h-cobordism theorem to recognize that the end of $W'$ is homeomorphic to $S^3 \times (1,\infty)$, from which you see that you can compactify $W'$ to a manifold by adding in a point.
In the case at hand, you can be more explicit. One way (this is what happens in Freedman's original paper) is to consider the building block $P \times I$, and then do (spin) surgery on a circle to kill the fundamental group, resulting in a compact manifold $P'$. The embedding theorems of Freedman find a (topological, locally flat) wedge of spheres in $P'$, which can be surgered out to give a compact, simply-connected homology cobordism from $P$ to itself. Stacking infinitely many of these gives $W'$. Presumably, although I've never done this, you can use techniques of Bizaca to build a handlebody picture of $W'$ from this description.
An alternate approach, which might be more amenable to drawing pictures, would come from Freedman's older paper, A Fake $S^3 \times R$. In this paper, which precedes his disk embedding theorem (but has many of the basic ideas, including reimbedding techniques) he constructs what I've called $W'$ by embedding Casson handles. There is also a Bourbaki exposition of this paper by Siebenmann that is helpful in trying to read it.