I recently read something along the lines of "General topology deals with the nice properties of pathological spaces, and algebraic topology with the pathological properties of nice spaces".
In a sense, this is consistent with my experience of both subjects, at least the part concerning which spaces we're talking about : in general topology often one thinks of the worst possible cases (e.g. totally disconnected spaces, non separated spaces, etc.), but in algebraic topology those are usually kept out : if someone says "but the space has to be connected", in algebraic topology we'll gladly add this as a hypothesis without a second thought; often we deal only with CW-complexes, or "connected, locally path-connected, semi-locally simply connected spaces" which are very far from the beasts one is concerned about in general topology.
Of course this is very schematic and I'm just generalizing stuff (though if you think this is a bad generalization, please tell me so !), especially about general topology.
However it's easy to come up with natural statements concerning nice spaces in general topology, we can easily add hypotheses such as path-connectedness etc.
But can we go in the other direction ? Is there some "algebraic topology of pathological spaces" ?
Of course it would have to look very different from what algebraic topology usually is (at least the one I know) : for instance the fundamental group of a totally disconnected space is quite useless.
But I can't see a reason why, a priori, one couldn't associate interesting algebraic invariants to pathological spaces, perhaps they wouldn't be groups, but some other thing. For a more formal question (though still vague) : is there a reason why there aren't nice functors from $\mathbf{Top}$ to "algebraic" categories ?
(I'm adding the "category theory" tag because there might be an answer concerning the structure of $\mathbf{Top}$ that would hinder the existence of such functors)
Best Answer
Actually, there is an active theory of algebraic topology for "pathological" spaces that has come a long way in the past two decades: Wild (algebraic/geometric) Topology. In fact, this has become a small field in its own right with a lot of recent momentum.
Descriptions of fundamental groups do become more complicated because in a wild space there may be shrinking sequences of non-trivial loops, which allow you to form various kinds of infinite products in $\pi_1$. Hence wild algebraic topology requires more than just the usual tools from algebraic topology but also is deeply connected to linear order theory, continuum theory, descriptive set theory, and topological algebra.
Here is an example of an astonishing result from this field, which addresses your interest in detecting homotopy type:
Homotopy Classification of 1-Dimensional Peano Continua (K. Eda): Two 1-dimensional Peano continua (e.g. Hawaiian earring, Sierpinski carpet/triangle, Menger cuber) are homotopy equivalent if and only if their fundamental groups are isomorphic.
The combined work of Greg Conner and Curtis Kent announced last year proves the same thing is true for planar Peano continua.
Once you realize how complicated these groups are due to the kinds of infinite products that can occur (although a word calculus of sorts does exist), it is absolutely remarkable that such theorems are true...almost scandalous. Results like the one above are very hard to prove. Eda's result required a lot of ingenuity and machinery that is being used and extended in current work.
Here is a little more pre-2000 history:
1950s - 1960s: There were a few scattered papers by some prominent mathematicians, e.g. Barrat/Milnor, H.B. Griffiths, Curtis/Fort.
1970s: Shape theory was developed to extend homotopy theoretic methods to provide invariants for more general spaces. The idea of space theory is to understand objects as (or at least approximated by) inverse limits of the usual "nice" spaces, applying your invariant to the nice approximating spaces, and call the inverse system of algebraic objects a "pro-invariant" and the inverse limit a "shape-invariant." The book Shape Theory by Segal and Mardesic is, I think, the best book on this topic. However, shape invariants only sometimes help with understanding homotopy type and traditional algebraic invariants of wild spaces.
1980s: Not much happened except for Morgan and Morrison fixing H.B. Griffiths description of the fundamental group of the Hawaiian earring.
1990s: Katsuya Eda, whose background was in logic, discovered that the Fundamental group of the Hawaiian earring behaves like a non-abelian version of the famous Specker group $\prod_{\mathbb{N}}\mathbb{Z}$. Eda was the first to make the key connection to order theory and describe the Hawaiian earring group as a group of reduced linear words $w$ (like a free group) where $w$ has countably many letters and each letter of your alphabet can only appear finitely many times in $w$. This work made the Hawaiian earring group practical to use; it is the key to many recent advancements.
Since Eda's work there has been a great deal done and there is now a huge amount of literature on the subject.