You may have a look at http://mathoverflow.net/questions/80777/what-is-a-continuous-path for a very much related discussion.

My opinion, in two words, is that the main property of $[0,1]$ is that one can glue the intervals and obtain basically the same thing. I think that one can define the fundamental group as soon as one has a way to describe paths using an ordered set for which the concatenation of intervals is well-behaved. Let me give an explicit example: in A-homotopy theory of graphs, one uses the natural numbers as ordered set and defines the continuous paths to be finitely supported functions from $\mathbb N$ to the graph with the property that $f(i-1)$ is a neighbor of $f(i)$. Then defines homotopy equivalence and gets a group that has several good properties. If you are interested, you may find details in Chapter 2 of http://arxiv.org/abs/1111.0268

I never really went through the details but I am confident that one can do something similar also using the hyperreal numbers as ordered set and define for instance the fundamental group of *$\mathbb R^2$ and *$\mathbb R^3$. I think that there is some hope to prove that they are not homeomorphic. This fact does not seem easy to prove using classical notions, as you can see here http://mathoverflow.net/questions/86562/non-standard-algebraic-topology