Check out 250.66(A) which states that the sole connection from the service bond to the first ground rod shall not be required to be larger than 6 AWG copper or 4 AWG aluminum. Now look at 250.53(C) which requires the conductor connecting multiple ground rods to be sized in accordance with 250.66 which requires that you use table 250.66.
Now for example, if the size of the largest service entrance conductor is between 350 - 600 kcmil then you are required to use a 1/0 AWG grounding electrode conductor everywhere except for at the locations described in 250.66(A) thru (C). As stated previously, 250.66(A) allows for a maximum 6 AWG copper to be used for the sole connection from the service bond to the first ground rod where table 250.66 requires a 1/0 AWG for the conductor connecting multiple ground rods.
There are similar exceptions for sizing the conductor that is the sole connection from the service bond to the ground ring and/or concrete encased electrodes. If you wire the ground ring and ground rods in series you could easily have different size conductors for the interconnection of the various required electrodes.
I'm not saying it makes sense, just interpreting what the NEC requires.