Have there been (successful) attempts to use something other than spheres for homotopy groups? Homotopy groups are famous invariants in algebraic topology. They have a myriad of wonderful properties:


*

*For $n \ge 1$, $\pi_n(X,*)$ is a group; for $n \ge 2$, this group is abelian.

*$\pi_n$ defines a functor from based spaces to (abelian) groups.

*$\pi_n$ is invariant under homotopy equivalences, and homotopic maps induce the same morphism on homotopy groups.

*The Seifert–van Kampen theorem allows one to compute the fundamental group of $X \cup_A Y$.

*A fibration $F \to E \to B$ yields a long exact sequence of homotopy groups.

*There is a natural suspension morphism $\pi_n(X) \to \pi_{n+1}(\Sigma X)$, and it eventually stabilizes as described by the Freudenthal suspension theorem.

*If $f : X \to Y$ is a map between two CW-complex that induces an isomorphism on all homotopy groups, then it's a homotopy equivalence.

*The Whitehead product endows $\pi_*(X)$ with a shifted Lie algebra structure.

*Etc., etc.


But after all, $\pi_n(X,*)$ is "merely" the set of morphisms in the homotopy category of pointed spaces from the $n$-sphere to $X$: $\pi_n(X,*) = \hom_{\mathsf{hTop}_*}((S^n,*), (X, *))$. This has always seemed somewhat "biased" to me; spheres are given a preponderant role in the definition. They are certainly a fundamental type of spaces, but they're not all there is to life. So my question is:

Have there been (successful) attempts to use something other than spheres for homotopy groups, in a way that some part of the structure described above is retained (in one form or the other)?

For example, maybe I'm interested in totally disconnected spaces, and I'd consider something like $\{ [0,1]^n \cap \mathbb{Q} \to X \}$ (with boundary conditions) up to "homotopy". And even here we see spheres (or at least Euclidean spaces) lurking in the background: the definition of a homotopy involves a path $[0,1] \to Y^X$, so why not consider "homotopies" of the type $[0,1] \cap \mathbb{Q} \to Y^X$...? One problem that could crop up here is that $[0,1] \cap \mathbb{Q}$ is not compact, so a lot of the theory goes out of the window.
Or maybe I'm interested in what happens in infinite dimension, so why not replace the spheres with some compactification of an infinite dimensional Banach space, for example? I have no idea what such a thing would look like.

So far, I've found three things in this direction:


*

*Homotopy groups with coefficients replace the sphere with a Moore space $M(G,n)$ (i.e. a space with a single nonzero reduced homology group). A sphere is in particular a Moore space of type $M(\mathbb{Z},n)$. Unfortunately such Moore spaces are not, in general, co-H-spaces, so $[M(G,n), X]$ is not a group, merely a pointed set. One reference for this seems to be Weibel K-book, but as far as I can tell they are only used as motivation for the definition of K-theory with coefficients (I haven't had the chance to read the book in detail).

*Hazewinkel, in Encyclopaedia of Mathematics, mentions something called "toroidal homotopy groups". I haven't been able to find much information about it anywhere and I can only guess it has something to do with tori. (I don't have access to the book, only to a few pages on Google Books (and other people may not even be able to see the same pages as I do))

*Last week I saw this preprint about "big fundamental groups" (the interval is apparently replaced with an interval of large cardinality) in the daily arXiv email, which I think pushed me towards this question.

 A: I can give specific information on the "toroidal" question, which I expect refers to:
Homotopy Groups and Torus Homotopy Groups
Ralph H. Fox
Annals of Mathematics
Second Series, Vol. 49, No. 2 (Apr., 1948), pp. 471-510.
I heard ages ago from Brian Griffiths that Fox was hoping for, but did not succeed in getting, a higher version of the van Kampen Theorem. 
Such versions are available, and involve cubes rather than spheres; they do not calculate directly homotopy groups, but some homotopy $n$-types, and moreover they work on certain "structured spaces", either filtered spaces, or $n$-cubes of spaces. There are presentations on my preprint page (...., Paris, Galway, Aveiro)  if you want more information. 
To comment on one of your initial bullet points, the work referred to in the last paragraph started in 1965 with disappointment  that the standard Seifert-van Kampen theorem for the fundamental group of based spaces did not yield the fundamental group of the circle, THE basic example in algebraic topology. That was just the start of computations of fundamental groups with which that theorem did not cope.  It turned out one needed more than one base point, hence groupoids.  See this mathoverflow discussion on using more than one base point. 
Nov 5, 2016 Here is a recent article on Modelling and Computing Homotopy Types: I developing part of the Aveiro talk. 
A: 
But after all, $\pi_n(X,*)$ is "merely" the set of morphisms in the homotopy category of pointed spaces from the $n$-sphere to $X$: $\pi_n(X,*) = \hom_{\mathsf{hTop}_*}((S^n,*), (X, *))$. This has always seemed somewhat "biased" to me; spheres are given a preponderant role in the definition.

Spheres emerge naturally out of the relationship between homotopy theory and higher category theory. One example of this is that you might object that the study of $\pi_1$ somehow arbitrarily picks out $S^1$, but in fact you can define $\pi_1$ without mentioning $S^1$ at all, using the theory of covering spaces or locally constant sheaves. One definition is the following: $\pi_1(X, x)$ is the automorphism group of the functor given by taking the stalk of a locally constant sheaf on $X$ at $x$. This is a functor on the homotopy category of reasonable pointed spaces and the circle naturally emerges as the object representing this functor. 
Similarly you can describe the fundamental $n$-groupoid $\Pi_n(X)$ of $X$ (which knows about the first $n$ homotopy groups) without mentioning paths at all, using a theory of "higher" locally constant sheaves (locally constant sheaves of $(n-1)$-groupoids, rather than sets). 
If you believe one way or another that fundamental $n$-groupoids are reasonable things to study (inductively: if you believe that homotopies are reasonable, then you believe that homotopies between homotopies are reasonable, etc.; this means you care about the interval $[0, 1]$ but it doesn't yet obligate you to care about spheres), then the higher homotopy groups naturally emerge by repeatedly taking automorphisms:


*

*$\pi_1(X, x)$ is the automorphism group of $x$ as an object in the fundamental groupoid $\Pi_1(X)$,

*$\pi_2(X, x)$ is the automorphism group of the identity path $x \to x$ as an morphism in the fundamental $2$-groupoid $\Pi_2(X)$,


etc. None of this explicitly requires talking about spheres, but again, these are all very natural functors on pointed homotopy types and spheres represent them. 
Said another way, among homotopy types, the $n$-sphere $S^n$ has a universal property: it is the free homotopy type on an automorphism of an automorphism of... of a point. These aren't just arbitrary spaces we picked because we like them; they're fundamental to homotopy theory in the same way that $\mathbb{Z}$ is fundamental to group theory. 
A: The "big" homotopy theory was first defined/explored in a paper J. Cannon and G. Conner wrote several years ago.  It was simply a tool constructed to understand a sort of generalization of the Hawaiian Earring (and its fundamental group).  
Consider the one-point compactification of uncountably many open (real) intervals.  The fundamental group of this space is too restrictive, only allowing a loop to traverse countably many loops, whereas the big fundamental group has no such restriction.  So its advantage over the fundamental group is merely a counting problem.  When you're trying to detect holes in spaces which have too high cardinality, or too many points packed in, or something like that.  
Consider the long line, with the ends identified--a path cannot make it all the way around, but a "big path" can.  Problems like this arise when you have "huge" spaces, and the big fundamental group is a sort of workaround.
A: One can indeed replace the sphere $S^n$ in the definition of homotopy groups $\pi_n(X,*)=[S^n,X]_*$ by any pointed space $A$, giving the $A$-homotopy groups $\pi_n(X;A)=[\Sigma^n A, X]_*$. As it happens, these $A$-homotopy groups are actually just the (usual) homotopy groups of the mapping space $Map(A,X)$:$$\begin{aligned}
\pi_n(X;A)=[\Sigma^n A, X]_* &\cong [A\wedge S^n, X]_* \\
&\cong [S^n, Map(A,X)]_* \\
&\cong \pi_n (Map(A,X), \text{const})
\end{aligned}.$$
Many tools of homotopy theory can be generalised to the $A$-cellular theory: there are $A$-cellular complexes, obtained by inductively gluing $n$-dimensional $A$-cells $\Sigma^nA \hookrightarrow C\Sigma^nA$ (or taking homotopy cofibers of maps from $\Sigma^n A$), and an $A$-Whitehead theorem : a map between $A$-cellular spaces that induces isomorphisms on all $A$-homotopy groups is a weak equivalence.

This material can be found in Dror Farjoun's 'Cellular spaces, null spaces and homotopy localization'.
