Can a linear transformation have exactly one invariant line? Is it possible for a linear transformation to have exactly one invariant line? My instinct is that it is not, but I'm not sure how to go about proving it. I'm interested in the two- or- three-dimensional cases, although it would be interesting if this is something that changes at high dimensions.

Yes, this would correspond to a matrix with only one eigenvector. Example: $f(x)=Ax$ with $A=\begin{bmatrix} 2 & 1\\\ 0 & 2 \end{bmatrix}$ or $A=\begin{bmatrix} 2 & 1 & 0 \\\ 0 & 2 & 1 \\\ 0 & 0 & 2 \end{bmatrix}$. The direction of the invariant line passing through the origin is $v=(1,0)^T$ and $v=(1,0,0)^T$, resp (that's the eigenvectors).