A finite-dimensional associative $\mathbf{k}$-algebra $\mathbf{k}Q/I$ is of tame representation type if for each dimension vector $d\geq 0$, with the exception of maybe finitely many ~~dimension vectors $d$ ~~ representations^{*}, the indecomposable representations of $\mathbf{k}Q/I$ with that dimension vector can be described up to isomorphism as finitely many one-parameter families, the parameter coming from $\mathbf{k}$.

What is an illustrative example of a tame algebra? Specifically, what's an example of a quiver $Q$ and admissible ideal $I$ such that (1) for some dimension vectors $d$ the indecomposables can't be described as finitely many one-parameter families^{*}, and (2) for some dimension vectors there is more than just one one-parameter family.

I ask because my go-to example of a tame algebra now is the path algebra of the Jordan quiver, the quiver having one vertex and one loop, over an algebraically closed field. But this example doesn't utilize all the wiggle-room that the definition of a tame algebra allows. So I'm hoping there is a better quintessential example to keep in mind.

***** Note that, as it was originally written, this was not the correct definition of an algebra having tame representation type, and actually condition (1) is not possible. See the comments below for the correct definition.

per dimension vector, just as you say. And according to what Hugh Thomas is saying below here and here, this isnotequivalent to the correct definition. $\endgroup$ – Mike Pierce Feb 17 '20 at 16:55