What's an illustrative example of a tame algebra?

I think the following is an example of a tame algebra where there is more than one component to a moduli space of fixed dimension. I don't know any examples where there are dimension vectors with moduli of dimension $>1$.

Take a quiver with two vertices $1$ and $2$, two arrows $x_1$ and $x_2$ from $1$ to $2$ and two arrows $y_1$ and $y_2$ from $2$ to $1$. Impose the relations $x_i y_j = 0$ and $y_j x_i = 0$ for $1 \leq i,j \leq 2$. I believe every indecomposable representation either satisfies $x_1=x_2=0$ or $y_1=y_2=0$. Thus, every indecomposable representation is a representation of either the Kronecker quiver $1 \rightrightarrows 2$ or else $1 \leftleftarrows 2$, both of which are tame, so this is tame.

For each dimension vector of the form $(n,n)$, we get two families of indecomposable representations, coming from choosing whether to make $x_1=x_2=0$ or else $y_1=y_2=0$.

Any quiver whose graph is affine Dynkin graph $\tilde{D}_4$, $I=0$. If all arrows look at the center, this is related to the 4-subspace problem, which is tame.

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.