This is not nearly as nice an example as the others, but I always imagined the line in characteristic five geometry as a countable set of points that glow like blue Christmas tree lights vaguely in the shape of a narrow paraboloid, with 0 at the vertex, with 1, 2,3, and 4 at the next "height", with the quadratic "irrationals" next, etc. although the makes it seem like each field of order $p^n$ contains the field of order $p^{n-1}$. Like in Carnahan's answer, I imagine the Frobenius automorphism shuffling around everything in each fixed ring. Finally, I imagine the Zariski topology as a glowy light filling in the paraboloid representing the forces that each point exerts on all others; it's constant everywhere, as in the Zariski topology, there's no real sense of distance.
This is not nearly as nice an example as the others, but I always imagined the line in characteristic five geometry as a countable set of points that glow like blue Christmas tree lights vaguely in the shape of a narrow paraboloid, with 0 at the vertex, with 1, 2,3, and 4 at the next "height", with the quadratic "irrationals" next, etc. although the makes it seem like each field of order $p^n$ contains the field of order $p^{n-1}$. Like in Carnahan's answer, I imagine the Frobenius automorphism shuffling around everything in each fixed ring. Finally, I imagine the Zariski topology as a glowy light filling in the paraboloid representing the forces that each point exerts on all others; it's constant everywhere, as in the Zariski topology, there's no real sense of distance.