Coming from a background of mostly algebra and geometry, I am curious to learn what kinds of spaces one can build using CW complexes. To put it bluntly, my question is:
Which "geometric" category is the largest one can build (all/some/most of) the topological spaces of using CW complexes?
The Wikipedia page lists several examples here, however a wider perspective on the landscape of possibilities would be nice.
It seems clear that not all topological spaces are CW complexes: requiring that the space be Hausdorff eliminates many "pathological" examples (e.g. the Hawaiian earring), but also many spaces of interest (e.g. spaces with Zariski topology).
On the positive side, polyhedra are , and most nice manifolds are (homotopy equivalent to) CW complexes (see here). Moreover, as per the Wikipedia page, real and complex algebraic varieties (using their Euclidean topologies I suppose) are CW complexes. I am also suspecting that the kinds of stratified spaces studied in Intersection Homology (topological pseudomanifolds?) are good candidates.
Perhaps my geometric view is also too constrained, any kinds of CW spaces that arise in analysis are also welcome.
It depends on your perspective, and in a lot of ways its a difficult question to answer.
For example, I study algebraic topology. What is important for me is that, as remarked in the comments, every topological space is weakly homotopy equivalent to a CW complex. In some sense (and with an appropriate amount of hand-waving) the category of CW complexes is the "correct setting" for doing homotopy theory. This is a possible answer to your initial question.
However, from another standpoint, it's not the correct setting for anything, and the standpoint in question is dependent on what you mean by "geometric category". Are you an algebraic geometer? A differential geometer? An analyst? Are you sure you only want to restrict to topological spaces? Need they be metrizable?
The point that I'm trying to get across is that you probably want to (and really need to) have an appropriate notion of "sameness" in mind when you ask questions along the lines of "what spaces are CW complexes?". What you're really asking is "What spaces are the same as CW complexes?". Homotopy equivalent? Homeomorphic? Diffeomorphic? Isometric? Equal? This is more than just a philosophical point.
Again, as remarked in the comments, the "largest category which can be built from CW complexes" is the category of CW complexes. The objects are CW complexes, and the maps between them are maps of CW complexes. There are a multitude of answers to these questions available in lots of places. One which hasn't been mentioned in the comments - and which is analytical in nature - is an infinite dimensional Hilbert space. This isn't a CW complex. Differentiable manifolds have the homotopy type of CW complexes. The Hawaiian earring does not.
EDIT: In response to your comment, an example of a CW complex which is not a manifold is (for example) $S^1 \vee S^1$.