The word or phrase to use depends on context. Inception, as a word, has nothing to do with nested objects. As a movie title, however, that is what it makes people think about it, especially if the thing you're talking about is confusing. In that sense, you'd describe the structure by comparing it to the movie Inception, rather than saying it is an inception.
When people make this comparison, they are usually trying to be funny (though I feel like, after a few years, the humor has started to wear off). This may not always refer to "things within things," though. Often, this is just used to describe something complicated, or something with an unexpected "plot twist."
A: I'm writing a story about a person struggling to write a story about her parents, who are struggling writers.
B: Sounds like Inception. BWOOOONNNG.
An adjective that may fit your need is meta -- it's a colloquial word derived from the prefix, and is used when something refers to itself.
Psychology is thinking about how people think. As a Psychology professor, I have to think about how people think about how people think. It's pretty meta.
In computer science, the term is recursion, and you would describe such a structure as recursive. This is definitely the appropriate formal term for it. However, the term may only be well known by computer scientists, mathematicians, and programmers. The average person, especially if they're not particularly familiar with computers, may not really understand this word.
In a more informal context, a lot of people are familiar with Russian nesting dolls, in which dolls are contained within dolls within dolls. If you made a comparison with these dolls, your meaning would probably be clear.
The government is like a Russian nesting doll of bureaucracy. Bureaus overseeing agencies overseeing departments.
Most generically, they would be called lapels (the left lapel and the right lapel). And, to the best of my knowledge, it's absolutely irrelevant whether it's a coat or a bathrobe that you're talking about—they're still called lapels. There are different types of lapels such as notch lapels, peak lapels and so on and so forth, but they're all lapels.
Best Answer
There are several terms used, depending on the size, number and where you live.
If there are lots of small ones (grapes, apples, oranges): if you are American, you would call them seeds and, if you are British, you would call small ones seeds and slightly bigger ones (especially citrus fruit) pips.
If there is just one large one (olives, cherries, peaches etc): if you are American, you would call it a pit and, if you are British, you would call it a stone.
You can use this NGRAM graph to experiment with British/American terms for different fruits. As the NGram graph shows, these are not hard and fast rules.
One notable exception is processed fruits like dates and olives. Olives are mainly grown, prepared and packed in non-English speaking countries. They supply to both US (141,000 tonnes per year) and UK (1,600 tonnes per year). It's easy to see from these figures why the suppliers choose to use the American term "Pitted Olives" on their packaging, even on products supplied to the UK. The import figures were obtained from here and here.