Conglomeration for conglomeration's sake I know that  category theorists have stretched the ontology of collections into conglomerates, 2-conglomerates, etc. My question is how far have they taken this? Are they interested at all in taking this concept as far as it can go, or is it really just a tool for "normal" maths to them? If no one is interested, why not?
 A: So this is not a complete answer, but it is an example that I think should make a hierarchy of super-proper classes less interesting. 
For any theory $T$ of sets, without proper classes, you can expand it to a theory $T^*$ with proper classes and impredicative class comprehension such that $T^*$ is equiconsistent with with $T$. The proof is due to Hao Wang where historically $T=\mathrm{NF}$ and $T^*=\mathrm{ML}$. If you do this with $T=\mathrm{ZF}$ then $T^*$ is something almost sorta like MK with semisets, which is poorly behaved, so we strengthen $T^*$ with full separation for sets and get, in essence, MK. Now we know MK is roughly equivalent to ZF plus some large cardinal assumptions (which basically give us a Grothendieck universe), so we may as well just use this latter.
The picture I'm trying to draw is this: we can add useless extra layers of classes to an existing theory to no particular benefit, producing $T^*$; or we can strengthen $T^*$ but still have just as much confidence in any strengthening of $T$ that's mutually interpretable with our strengthened $T^*$. 
Proof-theoretically, NBG is nice because it's finitely axiomatizable; MK does not share this feature, and I would be very startled if Adamek et al.'s theory did either. Barring this, the only reason I'm aware of to want proper classes and conglomerates is some idea of "naturalness". And while proper classes can (in most treatments) be described solely with $\in$ as those classes that are not $\in$ anything, conglomerates and higher tend to need extra predicates in the language, so it becomes sloppy rather quickly.
Hopefully I've not made any serious errors above; NF/ML are more my area, so I'm largely analogizing using my vague knowledge of ZF relatives. I hope that I am corrected if I have made any serious errors.
A: It seems like we can define a power "collection" operator, compose it transfinitely, and apply it to a proper class to expand this notion. Then if we admit class sized ordinals, the fun really takes off! If this attempt fails though, it'd still be useful to document other attempts, if any are out there.
