Quiver with r loops ($L_r$)

This quiver has one vertex and $r$ loops. Paths are words in the loops, so its path algebra is a free associative algebra. A representation assigns an arbitrary endomorphism to each loop.

Its path algebra is $k[t]$. Finite-dimensional representations are vector spaces with a single endomorphism. Thus oriented cycles can give an infinite-dimensional algebra even when the representations studied are finite-dimensional.