I previously noted that there are cases in which we say that something is possible (or use another modality definable in terms of possibility) for which possible worlds semantics seems to provide no adequate translation. I called these, for lack of a better term, 'superpossibles'. The two obvious cases are the actual world and God. And I also noted that this raised the question of what other cases there might be.
One candidate would be causal powers, and for reasons analogous to those seen with the actual world and God: namely, that possible worlds talk has no adequate way to handle either actuality itself or hierarchies and dependencies among possibilities. We can run the argument either by linking causal powers to the actual world or to God. First, the reason the modalities of the actual world can't be adequately captured by possible worlds talk is that the actual world itself has different possibilities, and therefore cannot be identified with a possible world, but also cannot be reduced to either a collection of possible worlds or an element in a possible world. But if one accepts the existence of causal powers, it is the causal powers that make the actual world have different possibilities in the first place. Therefore, a fortiori, causal powers deal with possibilities that can, at best, only partially be translated into talk about possible worlds, and this means that causal powers are superpossibles. Or one can analogize to God: modalities attributable to God can't adequately be translated into possible worlds talk because the latter treats possibilities all on the level; but when one attributes possibility to (say) God's existence, one does so in a way that makes this false -- God's possibility has a priority or fundamentality with respect to other possibilities that possible worlds talk has no way to capture. God's possibility is connected to His being able to make other possibilities possible. But causal powers are also actuality-makers, and have a fundamentality with respect to other possibles, although on a smaller scale. So similar reasoning applies.
Because possible worlds semantics is just a device for relating possibilities in an extensional way, so you can consistently assign truth values to modal claims without making radical changes to the logic you use, it treats possibilities as nothing more than instances (in a possible world) and necessities as nothing more than regularities (across possible worlds). But not all our uses of modality work this way, so that possible worlds talk flattens out our notions of modality. If something would be describable in possible worlds analysis as 'true in all possible worlds' or 'true in all possible worlds with such-and-such feature), this could be because it is made to have this feature by something else or because of its own nature; and, likewise, when we talk about possibilities we can perfectly well treat possibilities as having hierarchical relations of dependency between each other, rather than treating them all on a level.
I had noted previously that one of the reasons for the use of the possible worlds framework is that it is, with a few tricks, easily adaptable to talking about a lot of different modalities. All of the candidates for superpossibles that I've mentioned arise with regard to alethic modalities themselves (possibility, necessity); but there are non-alethic modalities that also sometimes get discussed in terms of possible worlds. One would expect superpossibles in these cases, as well. And we can identify a few obvious candidates. Take, for instance, Kant's categorical imperative in terms of the modalities of obligation and permissibility (deontic necessity and possibility). The categorical imperative, by its very definition, has a fundamentality with regard to other permissible things -- whether other things are permissible depends on the categorical imperative, so that the categorical imperative does not involve permissibility in the way other imperatives do, even though obviously it is permissible to do what the categorical imperative requires. So the categorical imperative is a plausible candidate for a deontic superpossible, a superpermissible, whose modal character cannot adequately be captured by possible worlds semantics. A similar argument can be run for regarding self-evident first principles as candidates for epistemic superpossibles.
In other words, the existence of superpossibles, that is, of things whose modalities can only, at best, be partially captured by possible worlds analysis, seems to arise from (1) the limitations in how possible worlds analysis can handle actuality; and (2) the fact that possible worlds analysis assumes that 'possibility' is univocal. It has difficulty, then, capturing modalities intimately linked with actuality, and with modalities that are analogically or hierarchically related, according to prior and posterior. This raises the question of whether there are other lines of argument by which one could discover superpossibles.