What kind of sphere would this be? I am new to topology, so please forgive any lack of precision. Would an n-dimensional object that had the interior of a sphere (e.g. the set of all "diameters" through the origin), but the end points of all the diameters end in the same single point, be considered exotic? If not then what would it be called?

I think you're looking for the quotient space $B^{n+1}/S^n$, where $B^{n+1}$ is the unit ball in $\mathbb{R}^{n+1}$ and $S^n$ is the unit sphere in $\mathbb{R}^{n+1}$. This space is actually homeomorphic to $S^{n+1}$! For a simple case, imagine pinching the boundary of the unit disc $B^2 \subseteq \mathbb{R}^2$ to a point, giving you $S^2$.
This is a sphere, but it doesn't really make sense to ask if it's exotic since we haven't given it a smooth structure.
