For the Kronecker quiver (two vertices, two arrows in the same direction) and dimension vector (1,1), over an algebraically closed ground field, the indecomposables are naturally parameterized by points in $\mathbb P^1(k)$. (The representation with the two maps given by $a$ and $b$ is sent to $[a:b]$.
For other tame quivers with no relations over an algebraically closed ground field, the situation is slightly worse: the natural indexing set for the representations whose dimension vector is the null root is $\mathbb P^1(k)$ with some points (up to three of them) counted more than once (but finitely many times). This happens in the example Bugs gave: there are three inhomogeneous tubes, each of width two, each containing two representations of dimension vector the null root, whereas the other points of $\mathbb P^1(k)$ each correspond to one representation. (With the all-inward orientation, the reason for the indexing by $\mathbb P^1(k)$ is that the moduli space of 4 points on $\mathbb P^1$—equivalent to representations with dimension vector $(1,1,1,1,2)$, i.e., the null root—is again $\mathbb P^1$.)
I am not quite sure what you mean by the extra wiggle room of type (1). Are these supposed to be dimension vectors that have only finitely many indecomposables? I would usually think that in that case, they can also be described by one-parameter families: just make the families constant.
