This question got me thinking about what makes the fundamental group (or groupoid) tick. What is so special about the circle? As another possible candidate for generalization, what about taking the one point compactification of the long closed ray R and thinking about homotopy theory with R in place of the interval? Would a theory of "long homotopy" arise? As a follow up, if this doesn't work, are there any other interesting instances of replacing the unit interval with another topological space and getting an interesting homotopy theory out of it? If not is there some characterization of the interval as the unique space which induces a nice homotopy theory?

The compactified long closed ray $\overline R$ will have two endpoints, but these are distinguishable. One has a neighbourhood homeomorphic to $[0,1)$ and the other doesn't. This scuppers "long homotopy" being a symmetric relation. (Also the transitivity would fail too.) The standard notion of homotopy relies on the interval $I$ having distinguished points $0$ and $1$, there being a self map of $I$ swapping $0$ and $1$, and there being a map from $I\coprod I/\sim$ to $I$ where $\sim$ is the equivalence relation identifying the $1$ in the first component to the $0$ in the second. These maps have to satisfy various formal properties. There is no continuous map of $\overline R$ swapping its "endpoints", so we can't mimic the classical notion of homotopy. 


Especially when doing topos theory, one sometimes uses the Sierpinski space (the twopoint space with one open point) as a sort of "directed interval." This is convenient because "Sierpinski homotopies" are exactly the 2cells in the 2category of topoi. For topological spaces regarded as (their sheaf) topoi, such 2cells are the pointwise ≤ relation in the specialization ordering. I think I recall that "geometric realization" relative to the Sierpinski interval is important too, perhaps it can be identified with some sort of descent. 


Perhaps the article of J. Cannon and G. Conner, "The big fundamental group, big Hawaiian earrings, and the big free groups", will interest you. I believe they work with just what you said, the onepoint compactification of the long closed ray, or something very similar. 

