What are examples of problems that are known to be sub-exponential, but
- are known to be non-polynomial, or
- are not known to be polynomial?
EDIT. Here is what I mean by sub-exponential (apologies for leaving this open). A function $f:\mathbb{N}\to\mathbb{N}$ is subexponential if $$f(n) = 2^{n^{o(1)}}.$$ (Thanks to Emil Jerabek for his help in the comment section below!)