The Taylor series and time complexity analysis
$2^n$ can be represented as a Taylor series, which is an infinite sum of polynomials. Notably, there is no single polynomial which is equal to $2^n$.

In general, since big-O lets us get rid of lower-degree terms in the polynomial, we can say that something has polynomial time (or space or whatever you're measuring) when it is $O(n^k)$ for some fixed $k$.

Since there's no single $k$ where $\forall n \ldotp O(2^n) = O(n^k)$, we say that $2^n$ is not polynomial.