Are quivers useful outside of Representation Theory?
In addition to being a nice example for abelian, $A_{\infty}$ and Calabi–Yau categories, and being a prototypical example for Generalized Donaldson–Thomas Invariants and the Wall Crossing Phenomenon, quivers have a lot of applications in various different fields. Since the question is asking for applications in addition to representation theory, I'm listing a few cases.
Most prominent is Algebraic Geometry, particularly Moduli problems and GIT (read motivations in Reineke's article) and Video lectures on quivers by Reineke at Newton Institute, Cambridge.
Recently, a correspondence has been proposed between Gromov–Witten invariants and Quivers. (Pandharipande–Gross).
Also in physics applications in String Theory, Supersymmetry, Black Holes and Particle physics.
Relation with quantum dilogarithm, number theory, and cluster algebras, read e.g. this review by Keller.
Also through the work of Nakajima, there is a relation to Instantons of Yang–Mills theory, Hitchin Moduli spaces and the theory of Hyperkähler manifolds.
As Pace Nielsen already posted, the strength of quiver theory is to provide easy examples and counterexamples.
The first applications are of course inside representation theory and ring theory, because Gabriel's Theorem states, that if you have a property of a finite dimensional algebra over an algebraically closed field that can be detected in the module category, then it suffices to look at path algebras of quivers (with relations). For example for proving that an algebra is wild, it suffices to find a subquiver (with relations) that is known to be wild; and there are several lists of such quivers. This is useful in representation theory of Lie algebras and finite groups.
There is the connection with Lie theory (and other things that can be classified via Dynkin diagrams) via the Hall algebra.
Cluster theory, which is an advanced topic in representation theory of quivers, has applications in geometry.
If you are given an algebra, I think it is a natural question is, whether it is possible to classify all the indecomposable representations. If this is possible you can work on a parametrization and understand better the module category by working with the classification. The tame-wild dichotomy helps you there, it answers the question if it is possible. If an algebra is wild, it is not possible to classify all representations. You have to ask other questions.
As was mentioned above, many moduli spaces have a quiver description; one of the most famous example is given by Nakajima quiver varieties, which are defined for any quiver (and they serve as the main example of symplectic complex varieties which are resolutions of an affine variety), but when the quiver is the affine quiver of ADE type, they describe moduli spaces of torsion free sheaves on the quotients ${\mathbb C}^2/\Gamma$ where $\Gamma$ is a finite subgroup of $SL(2)$ (these are also known as ALE spaces). These moduli spaces are very important in many places in mathematics and physics (gauge theory) and quiver description is very useful when you want to tackle some explicit problems related to them.