What can we call a set of sets of sets? When I’m referring to a set of sets, I like using the expression “a family of sets”, or “a collection of sets”, as is customary. But I was wondering, what if we needed to refer to a set of sets of sets? I have not found a specific word that identifies the third “layer”, as family and collection identify the second. Do you have any personal preference? I’d like to hear your suggestions.
 A: There is no benefit to bringing in a whole bunch of synonyms for the same concept.
Children, when first learning to communicate, are force-fed a whole bunch of more-or-less arbitrary rules by people whose thinking is generally not very sophisticated (or they'd be doing something more intellectual than baby-sitting a bunch of small children), and these rules get embedded.
One of these rules is something like: "When you are communicating something which requires the same concept to be used more than once, never use the same word for that concept." While this may or may not be a good rule to use when writing some flowery novel or other ephemeral entertainment, it is in general not a good rule to use in technical writing.
The main problem with saying something like "a collection of families of sets" is that the terms "collection" and "family" (most particularly "family") have more specialised meanings than merely being a synonym for "set". A "set" is defined precisely by means of a precise axiomatic framework which, at base, is that it is defined solely by the elements it contains, and has specific rules (for example: the Zermelo-Fraenkel axioms) by which it may be constructed in order to specifically exclude constructions which lead to inconsisencies and paradoxes. A "family" calls to mind the idea that there is an "indexing set" which seeks to identify each of the elements with the elements of an auxiliary set which is defined independently. So unless you really mean "family", it is recommended that you don't use it. On the other hand, "collection" is looser than "set". It has (as far as I know) no rigorously formal mathematical definition, and just means "a bunch of stuff".
In conclusion:
"A set of sets of sets" is perfectly adequate, if this is really what you mean to write about. Mind, you should consider the question that if you need to go three levels deep, you may need to think about a recursive definition for whatever it is you are seeking to define.
A: I am bringing definition of family from  Bourbaki Theory of sets: Function $f=(F,A,B)$ is defined by triple, where $A$, $B$ are sets, $F$ is functional graph and domain $pr_1F=A$. Functional graph can be called family, the domain is called index set and the range $pr_2F=B$ is called set of elements of family. Indicial notation $f_x$ is used to denote the value of $f$ at the element $x$.
