Complexity of not P or EXPTIME
A function $f(n)$ is time-constructible if there is a Turing machine that runs in time exactly $f(n)$, where $n$ is the input size (in bits). Kobayashi, in his paper On proving time constructibility of functions, showed that if $f(n)$ can be computed in time $O(f(n))$ then $f(n)$ is time-constructible. In particular, functions like $n^{\lfloor \log_2 n \rfloor}$ are time-constructible, even though they lie strictly between polynomial time and exponential time.

In fact, the time hierarchy theorem even shows that there are some decision problems which can be computed in time $O(n^{\log n})$ but not in polynomial time. This shows that there are problem not in $\mathsf{P}$ which can be computed in faster than exponential time.