I'm looking for a (possibly ontological) recognised term for a theory that acknowledges that things can be broken down into smaller things - essentially components, parts and eventually atoms, quarks etc. That effectively our world is a collection of collections. The term doesn't need to define whether the process of breaking down is infinite (or not) or that there is a commonality to things when they are broken down - just that they can be broken down is the commonality. The closest term I've found is Bundle Theory (https://en.wikipedia.org/wiki/Substance_theory#Bundle_theory) are there others?
I believe holarchy is what you're looking for. A holistic hierarchy. Rather than the great chain of being, it becomes the great nest of being. Nests within nests within nests. In this view, nothing is either whole or part, but rather whole/parts, simultaneously. E.g., an atom is a whole thing in and if itself, and quite remarkable. A molecule is a whole thing in and of itself, also. But it is made up of atoms, which are wholes acting as parts. A cell is similarly a whole, made up of molecules, made up of atoms. And the cell is even more remarkable. And it goes this way on to multicellular organelles, organisms, species, etc, ad infinitum. You can see this in other simple terms such as language; letters to words to sentences to paragraphs to poetry to stories to mythological cosmologies. Each part is also a whole and is fundamental to the entirety. Indeed, the "lower" you go, the more fundamental. While the "higher" you go, the more significant. If that seems incongruous, it still makes sense. In brief, a molecule is fundamental to my organism, and without molecules there would only be atoms raving about the universe. No more "me". However, this "me" is full of ideas, hopes, dreams, artistic impulses, morals, ethics... basically, "the good, the true, and the beautiful", ergo, more significant. So... a collection of collections is what we've got, and holarchy os the best name for it at present.
Yes, the idea is contained on the Theory of Systems.
The systems theory has solid basis. A systems is a group of parts working together. What are parts? just more systems (also subsystems). Therefore, systems are groups of groups... of groups, of groups... ad infinitum, or ad minimum. We haven't reached the minimum particle that can form a group, we usually call it atom, but it can be divided. It may not exist. We also don't know if there is a group that contains all. We used to call it universe, but maths grant the possibility of several, infinite universes. Anyway, in common words, a system has another name: thing. Then (only logical consequences):
A system is just a group. Therefore, a thing is just a group. Group parts are also systems (we use the word subsystems in order to identify the scale). Therefore the universe is a fractal structure, made by groups that hold groups recursively, and are part of higher level of existence groups (I didn't saw books concluding this, but this is just a logic conclusion, that I use on my writing).
Since here, I will express my opinion, since the systems theory is terribly incomplete, and most of my next book (previous one here: ) is dedicated to complementary stuff to it. You may want to read this opinion.
According to solipsism, everything exists only in our mind; I agree with that position, in spite solipsism is commonly rejected. We create systems in our mind probably in order to persist. We perceive the clouds, the air, the sea, but they are just groups of elementary particles, and... the same particles! We decide where do they end, and where do they begin (not my rational, but my embedded mechanisms). A finger is a system, a thing. Where does it begins and ends, depends on you. If you are instructed to cut someone's finger, you will probably do differently as me. Where does fire ends? you can put your hand there, and you can be wrong about it. Where is a society? you decide. You can name what you see a river, even if after ten seconds it is not there anymore. Your mug constantly changes (particles move and change places!). Your wife's cells -even opinions- are replaced periodically. But you know it's she. Groups-systems-things are created by our causal mechanisms of thinking and persisting.
Therefore, systems are just groups. Groups exist only in our minds. Physically, particles interact in fuzzy clouds. It is us -every animal, every entity- that give limits and even names to those clouds, in order to take profit (reducing internal entropy?) from them. End of opinion.
 http://www.merriam-webster.com/dictionary/system  http://ydor.org  http://www.newworldencyclopedia.org/entry/Solipsism  https://www.phy.duke.edu/~rgb/Philosophy/axioms/axioms/node43.html  Richard Feynman: "Everything is made of atoms", The Feynman Lectures on Physics, Volume I.
Maybe you are looking for a combination between mathematical fractals and Jungian archetypes.
As you know, the fractals are a structure repeating itself to infinite. So if we see the world as a collection of objects that are reapeating themselves with a little difference between them because it is not an "utopic" or "ideal" world, like time, space, events, movements, every object could be a collection.
The reference to Jung is made because everything that happens is written in another collection of collections in our collective mind.
I think this is appropiate for you, even if it is not a term that can be used.