[yorwba]

Something can be exponential in one context and polynomial in another. Asymptotic analysis is about the growth rate of a function in terms of some input variable. The results you get depend on which values you assume to be fixed while others vary. It is usually applied to analyze the runtime of a program in terms of the input size, which is the basis of classifying the runtime complexity, but that's not the only way you can do it.

If you have a sequence of programs with polynomial runtime, but which grows as N, N^2, N^3, N^4, ..., that's a textbook example of exponential growth. It is not generated by running a program on increasingly larger inputs, so there's nothing to be put in the exponential runtime complexity class, but it's exponential nonetheless.

[whatyoucantsay]

> If you have a sequence of programs with polynomial runtime, but which grows as N, N^2, N^3, N^4, ..., that's a textbook example of exponential growth

No, it isn't. N, N^2, N^3, N^4, ... is polynomial, not exponential. Exponential would be X^N. Look at the graph on the Wikipedia page I linked to.

[yorwba]

Each element of the sequence is a polynomial, but the sequence of polynomials grows exponentially. X^N is exponential in N, polynomial in X. N^X is exponential in X, polynomial in N.

[whatyoucantsay]

> Each element of the sequence is a polynomial, but the sequence of polynomials grows exponentially.

Bringing this back to the original comment about nesting loops, perhaps what you're describing is some sort of metaprogramming that dynamically generates increasingly deeply nested loops based upon the size of the input. If someone were to write that program then yes, it would be exponential.

In the far more common case of a programmer nesting a few loops, the resulting run-time would be polynomial, not exponential.

[IncRnd]

What you are describing now is called superpolynomial time in computer science.