The representation theory of the quiver with one vertex and two loops (that is, the problem of classifying finite dimensional representations of the free algebra $\mathbb{C} \langle x,y \rangle$ upto isomorphism) is 'in principle' accessible via the tools of geometric invariant theory.

First one fixes the dimension $n$ of the representation. Then the problem is trying to classify pairs of $n \times n$ matrices upto simultaneous conjugation, that is the orbits of $M_n(\mathbb{C}) \times M_n(\mathbb{C})$ under this action by $GL_n$.

The quotient variety $X_n = M_n(\mathbb{C}) \times M_n(\mathbb{C})/GL_n$ classifies the closed orbits which are the isomorphism classes of semi-simple representations. It is of dimension $n^2+1$ and its coordinate ring is generated by traces in words on $x$ and $y$. It can be stratified by locally closed smooth strata and the etale local structure in points $x \in S_{\tau}$ of such strata are given by similar quotient varieties of 'easier' (that is, lower dimensional) quiver-representations $(Q_{\tau},\alpha_{\tau})$.

To classify further all $n$-dimensional representations having as its Jordan-Holder semi-simplification the given point $x \in S_{\tau}$ amounts to studying the orbits in the nullcone of the quiver-setting $(Q_{\tau},\alpha_{\tau})$ and is best approached via the Hesselink-stratification, which again reduces to further quiver-representation problems (and vector-bundles over them).

An old account of this program is given in "Orbits of matrix tuples". Seminaires Congres, Societe Mathematique de France, 2:245–261, 1997
here's the preprint version : http://win.ua.ac.be/~lebruyn/LeBruyn1995c.pdf

When i said 'in principle' i mean that this program is only carried out in full detail for $n \leq 4$, and here's why:

Even for the generic stratum, corresponding to isomorphism classes of simple $n$-dimensional representations it is not known whether it is a rational variety (that is, whether one can parametrize simple orbits by a set of independent variables) unless $n \leq 4$ (by results of Procesi and Formanek).

For $n=5$ or more general for $n$ dividing $420 = 2^2.3.5.7$ we only know that the variety is stably rational and beyond those $n$ we do not know a thing. This is in an even older preprint: "Stable rationality of certain PGL(n) quotients". Inventiones Mathematicae, 104:179–199, 1991
http://win.ua.ac.be/~lebruyn/LeBruyn1989b.pdf

In recent years significant progress has been made in the study of moduli spaces of semi-stable quiver representations, look for instance on the arXiv for work by Markus Reineke. Aidan Schofield proved the remarkable fact that these moduli spaces are birational to the variety of tuples of $n \times n$
matrices under simultaneous conjugation where $n$ is the greatest common divisor of the components of the dimension vector, so eventually in all wild quiver-problems one bangs ones head against this most classical of all wild problems.