TL;DR: "Sub-subcategory" is not a thing.
In both language and taxonomy, there is no special word for the subcategory of another subcategory. There are several reasons for this.
For one, subcategory is a class of things, not a thing, itself. The prefix "sub-" indicates that all of the things in this class also belong to another, broader class of things. It does not infer how broad or narrow the parent class is.
So, grammatically, a subcategory of a subcategory is called... a subcategory.
If you have 15 levels of taxonomy, the 15th child category is still a subcategory. The 14th subcategory is both the child of the 13th subcategory and the parent of the 15th subcategory, and so on all the way up to the primary (or "main") category.
"Sub-sub-sub-(...)subcategory" is not only ugly, it literally makes no sense.
Which is perfect, because the more subcategories you make, the more confusing and bloated your taxonomy will be.
This is called diffuseness, and it defeats the point of categorising: to make sense of things.
So, if you can avoid more than two or three layers of category (primary/main category > parent subcategory > child subcategory), then you should.
It just makes sense.