Kleber discovered experimentally for the clockwise rotor sequence that after three million chips, Rn and rn differ by less than two. You can find here a chart showing Rn-rn for varying n depending on the chosen rotor sequence. The different colors represent different rotor sequences and the black lines are the moving averages. In fact, till n=3Mio, the maximum of Rn-rn is 1.21, 1.74, and 1.96 for ←↓↑→, ←↓→↑, and ←→↓↑, respectively. Note that (as it is common notation) the rotor sequences given here are "retrospective", not "prospective" rotors, that is, the rotor is updated first and then the chip is sent in the direction of the rotor. This way, the rotor says where the last particle went instead of where the next particle is going, which is sometimes advantageous. As noted by Jim Propp, the ordering of the Rn-rn in above figures exactly corresponds to the ordering of the local discrepancies of Holroyd and Propp.
As in above three examples, the final rotor direction can be used as a color to present the rotor-router aggregation. The resulting structures are very chaotic around the origin and somehow self-similar and fractal-like closer to the border. There is currently no good explanation where this structure comes from.
Step by step simulation of n chips takes time Θ(n2). Hence it is impossible to calculate large rotor-router aggregations with one or ten billion chips that way. In order to speed things up, Friedrich and Levine have presented an algorithm that computes the final state of the aggregation without computing all intermediate states. This takes approximately O(n log n) time. Their technique is based on a "least action principle" which characterizes the odometer function of the growth process. Starting from an educated guess for the odometer, they successively correct under- and overestimates and provably arrive at the correct final state. The source code is available here.