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.