The 1-sphere can certainly be built from a 0-cell and a 1-cell. Also from two 0-cells and two 1-cells. These are the canonical n-sphere structures as well.
Method 1: take an n-cell and a 0-cell.
Method 2: take 2 i-cells in each dimension with $0 \leq i \leq n$.
Of course there are infinitely many CW structures that describe any given $S^n$ with n finite. Countably infinite, I think, and this should be the case for the exhaustive list of any finite dimensional CW complex.
What I'm wondering is if this list can generally be described in closed-form.
For $S^1$, the only structures are, with $i>0$, i 0-cells and i 1-cells. Similarly for $D^1$, but with one less 1-cell.
Inductively, we can then describe an exhaustive list for the cell structure of $S^n$ by considering any number of $S^{n-1}$ (possibly with different cell structures) as 'latitudes' of $S^n$, which we can connect with $(n-1)$-cells and fill in with $n$-cells. Similarly for double attaching $n$-cells to copies of $D^n$. (We also need to include a 'degenerate' structures, like the $0$-cell/$n$-cell construction that isn't obtained inductively.)
So, although I don't want to write this out right now, all countably-many CW-complexes structures for a finite dimensional n-sphere can be enumerated by this process.
Is it generally possible to enumerate all the CW cell structures for an arbitrary finite-dimensional CW space? (up to the number of cells in each n-skeleton, ignore the attaching maps...)
Furthermore, what kind of information can be extracted from this list? There must be so much information contained in an exhaustive CW description of a space, even if we forget about the specifics of attaching for any particular cell structure. I really like the idea of describing topological structure with purely integer data.
You're not going to find an exhaustive list, because there are infinitely many possible CW-complexes homeomorphic to even $S^1$. (For any positive integer $n$, take as the 0-skeleton the $n$th roots of unity.)
A way to approach things is that a complex (either simplicial or CW) is a method of imposing a combinatorial structure on a topological space in such a way as to make extraction of homological information relatively straight forward. In the case of a simplicial complex, extracting the information is trivial, but the complex can be unwieldy (simplicial complexes behave poorly with respect to products, for example). CW complexes are meant to allow a minimal analogue, so trying to find complicated CW decompositions somewhat misses the point.