The van Kampen theorem implies that the fundamental group of the figure-eight, being the wedge product of two circles, is not abelian.
How can one prove directly that the path-product of the loop $\alpha$ around one of the two circles and the loop $\beta$ around the other (both at the point where the circles are joined) does depend on their order, i.e., that $\alpha \beta$ is not path-homotopic to $\beta \alpha$?
Surely there must be a way!