When I teach constant coefficient linear differential equations, the usual guess of an exponential can be motivated because it is "approximately" a fixed point for the differentiation operator. The aesthetic here being that mathematicians look at fixed points to exploit symmetry to find solutions.
If one considers a 2x2 first order matrix linear differential equation of the form $$X'=AX$$ when $A$ has a repeated eigenvalue with only a one-dimensional eigenspace, one can still discover what needs doing using the approximate fixed point idea while appealing to an analogy with the one-dimensional procedure of "reduction of order". This is done here, for example, in the way most of us probably teach it.
I always feel like I'm making a stretch when teaching this particular material, and was wondering if anyone had a more global explanation for the appearance of generalized eigenvectors here? I always felt that there should be, but can't think of a more natural way to explain this than the process found in the link above.