Structures all the way down
A structural property is a property that belongs to things in virtue of their constituents' properties and interrelations. For instance, the property being a methane molecule necessarily belongs to all and only things consisting of suitably connected carbon and hydrogen atoms.
There is two-way dependence: Necessarily, if something instantiates a structural property, then it has proper parts that instantiate certain other properties; conversely, if the proper parts of a thing instantiate those other properties then, necessarily, the thing itself instantiates the structural property.
It follows that structural properties can only belong to mereologically complex objects. Unless "property" is somehow restricted, the converse also holds, so that P is a structural property iff no mereological atom instantiates P. (Let P be a property that only applies to mereologically complex things. Let S be the set of those things. Then something is P iff all its parts have the property of being a part of a member of S. So P is structural. Here and from now on, all quantifiers are to be understood as possibilist quantifiers.)
We could, for example, restrict the part-properties to intrinsic properties. Let's say that P is an i-structural property if x has P whenever the relevant parts of x instantiate specific intrinsic properties and relations. (Yeah, that's terribly rough. Does anyone know a good and precise definition of structural properties that does not presuppose a special ontology of properties?) So no structural property that differs between things whose proper parts instantiate exactly the same intrinsic properties and relations is i-structural.
That a property is structural does not mean that we identify it structurally, in the sense that by definition or linguistic convention, "P" applies to things whose proper parts are such-and-such. We might, for example, use "methane" as a role term for whatever property -- simple or structural -- plays a certain causal or nomological role. Hence when "P" denotes a structural property, "P is structural" is often an a posteriori necessity.
The properties of a thing's constituents in virtue of which the thing itself has a certain structural property can themselves be structural: the constituents can have these properties in virtue of the properties of their own constituents. And these properties of the constituents' constituents can themselves be structural. Thus some properties might be 'structural all the way down'.
Not every infinitely structural property is structural all the way down in an interesting sense. Suppose every methane molecules ultimately consists of certain point-sized mereological atoms. (Actually, it doesn't matter whether the ultimate consituents are point-sized. But it helps the imagination.) So something is a methane molecule iff it has the right number of rightly arranged point-sized parts (of that kind). Still, methane molecules could have infinitely many parts of intermediary size: the molecules are made in part of carbon atoms, which are made in part of protons, which are made of quarks, and so on, without end. Then being a hydrogen atom would in a sense be infinitely structural despite being ultimately grounded in simple properties of mereological atoms.
We get a truly ungrounded structural property if we imagine away the basic constituents. Methane molecules are then bits of atomless gunk.
The relationship between consisting of gunk and having ungrounded structural properties is not entirely straightforward. For instance, it is not true that ungrounded structural properties can only belong to bits of gunk. They can also belong to atomic objects, as long as they are infinitely divisible. Suppose only some methane molecules have basic constituents. Then it is not true in general that something is a methane molecule iff it has such-and-such basic constituents, so being a methane molecule is not grounded in these simple properties.
Is every property that exclusively belongs to bits of atomless gunk an ungrounded structural property? I guess so, but that might depend on the exact definition of ungrounded structural properties. I'd say that, roughly, P is an ungrounded structural property if a) P is structural, and b) there are no simple (i.e. non-structural) properties (and relations) QQ such that x is P iff certain parts of x exhibit a certain pattern of instantiation of QQ. So consisting of atomless gunk probably counts as ungrounded.
(I meant to comment on an apparent change in Lewis's views on ungrounded structural properties here, but I got lost in the stage setting, so I have to postpone that for tomorrow. Apologies for this rather pointless entry.)