At the risk of missing your point and perhaps stating the obvious...
If an abstract object is that which cannot be physically (concretely) instantiated; if it is that which exists only in the imagination of those who contemplate it, then it is limited only by the limitations a person wishes to place upon it, and/or by the powers of a person's imagination.
Even a basic number such as the number 1, which is subject to shared understandings, can be reimagined to be constituted by a potentially infinite range of possible parts ('potentially infinite', because the imagination is likely unable to comprehensively conceive of an actual infinite or to actually imagine an infinite range of things).
A common imagining of the constituent parts of the number 1 might be the decimal fractions of the number 1, as these fractions or parts possess a utility we routinely employ.
There is nothing however to prevent a person imbuing the number 1 with any number of constituent parts which diverge from common usage (such as little pixel fairies, entire universes, abstract ideas, poetry...a combination of these), or to imagine a number 1 which has no constituent parts whatsoever. The number 1 can even be understood to mean something entirely different to what most people consider 1 to represent; including the number 2.6756 for instance, or an emotion, an instruction, a person, a rule, or a god.
Abstract objects are useful in part because they are malleable; because they can be adapted to suit a person's or a community's needs. If constituent parts of an abstract object prove of value to us, then those parts come into existence, and they don't have to conform to anyone else's notion of what those parts (if any) should be, unless the value they hold is shared and/or if their value must be translatable by the members of a group or by artificial systems.
In relation to your notion of a "collection of ideas that encapsulate it, 'make up' the idea"; ideas/parts that may constitute the number 1 might include concepts like 'quantity', 'relation' and 'sequence'. These in turn might have constituent parts, even if those parts are merely the arrangements of letters which allow us to articulate them, and to compare them with one another and combine them with one another.
Are there abstract concepts which can't be reduced? It seems unlikely if all abstract concepts can be constituted by the languages we use to express them, and if these languages can then be constituted by other properties, much in the same way the number 1 can be. This leads to a kind of circular constituency which could theoretically be repeated endlessly.