Assumptions. The overall claim is that, "X is possible," when X is some proposition, can be translated into, "If X is the object of a true conditional or disjunction, then the description possible applies to X." So for example, suppose that, "If the moon is made of green cheese, then space mice would be tempted to eat it," is a true conditional. But so then we would not (in this immediate context) refer to the antecedent as possible, but instead the consequent is an example of possibilia relative to the antecedent.
Unrestricted conditionality, then, would be necessity: "If [insert any stable proposition], then X," means that X obtains under all conditions whatsoever, i.e. is necessary.
As far as disjunction goes, we would say e.g., "If the moon is made of minerals or green cheese, then there is a possibility of the moon being made of green cheese (abstract though this possibility is)." I imagine that we would need to compromise on the principles that give us the standard explosion argument for the LNC, perhaps by adopting a relevance logic, to keep all this tidy enough; otherwise, we would end up with, "If a logical explosion occurred, then the LNC would be false," but we don't want to say that there is some sort of actual, though abstract, possibility of the LNC being false (at least not on account of the abstract possibility of a logical explosion!).
Questions. Supposing we can keep all that tidy enough, what happens if we iterate the modal operators (or predicates) so construed? I have a hazy idea in my head that we can correlate the "levels" of modality, with those iterates. So maybe, "If a logical explosion occurred..." goes with epistemic possibility, as the most ethereal, and one-off, iteration of the possibility descriptor. For present purposes I will conflate epistemic and logical modality; so let's say that something's being possibly possible (possible under a conditional/disjunction representing an anterior epistemic possibility?) is metaphysical possibility. Then possibly possible possibilities are nomological, or if we consider a force like free will as "the ability to do otherwise modulo a principle of alternative possibilities," a modal triplex can also map to the concept of free-will possibilia.
Now, I don't know that there are any substantive descriptions out there of levels of modality besides the epistemic/logical, the metaphysical, and the nomological/volitional. So I wouldn't know what to say of a fourfold iteration, and so on down the line, to say nothing of infinite iterations. But then I'm not sure any of these constructions, even as per the opening assumptions, actually do go through.