You may be interested in the classification of quivers of finite mutation-type:
http://www.emis.ams.org/journals/EJC/Volume_15/PDF/v15i1r139.pdf

This extends the classification of mutation classes of quivers determining cluster algebras with only finitely many cluster variables—the answer to this classification problem is given by the simply-laced Dynkin diagrams. (The non-simply-laced diagrams can also be included by defining cluster algebras from skew-symmetrizable, rather than just skew-symmetric, matrices.) These classes of quivers are sometimes said to have finite cluster-type.

All of these mutation classes are necessarily finite (one says the quivers have finite mutation-type), but there are also other quivers with finite mutation-type, including the affine Dynkin diagrams. (In type $\tilde{\mathsf{A}}$ one has to be a bit careful, because not all quivers with this underlying graph are mutation equivalent, and the cyclically oriented $\tilde{A}_n$ quiver is mutation equivalent to $\mathsf{D}_{n+1}$.) One might hope, given that the finite cluster-type problem was solved by Dynkin diagrams, that the slightly more general finite mutation-type problem might be solved by Dynkin + affine diagrams (cf. finite representation-type versus tame representation-type, or positive definite Tits forms versus positive semi-definite Tits forms). However, this is not the case, for several reasons of increasing severity.

Firstly a quiver on two vertices always has finite mutation-type for degeneracy reasons, but it doesn't feel too unreasonable to exclude this case. Next there are quivers determined by triangulations of oriented surfaces with marked points, which are also of finite mutation-type but their mutation classes needn't contain a Dynkin or affine diagram. Finally, Derksen and Weyman show in the paper linked to above that even excluding the previous two cases, there are still some exceptional mutation-finite quivers.