Let D(S) read as, "I disagree that S." It is possible to iterate this, so that DD(S), "I disagree that I disagree that S." Then we can go on to DDD(S), and so on. (For a peer-reviewed text that goes over its own theory of a disagreement operator, see Haret and Woltran, "Belief Revision Operators with Varying Attitudes Towards Initial Beliefs.")
At first, I was thinking that we would start losing track of the significance of iterated disagreement, like once we got to, say, D227(S) (for that many iterates), we would have trouble understanding what exactly it is that we are disagreeing with. Plainly enough, DD(S) already is not plausibly implied by just D(S), so we have a lack of a parallel with "standard" epistemic logic (where k(S) yields kk(S)) here. However, the disagreement operator can essentially figure in doxastic logic, which partly underlies/underscores epistemic logic (esp. when an epistemic logic carries a belief operator); and so its cognitive limitations would seem pertinent to the question of iterating the belief operator, and then even the knowledge operator itself.
On the other hand, I'm intuitively reading, "I disagree that I disagree that I disagree that S," as canceling out the denial of S-disagreement. Then D4(S) cancels that cancellation out, and so on. So maybe we can just flip back and forth between a pro-disagreement(!) moment in the unfolding expression, and an anti-disagreement one, rather than diffuse our cognition of our capacity for disagreement into the doxastic-modal ether.
Would the back-and-forth phenomenon occur in the iteration of the disagreement operator, or would the sense of disagreement break down eventually instead, yielding ramifications for the iteration of belief and knowledge operations in turn?