These would be problems in EXPTIME that are technically exponential, but are still practical. For example an algorithm that can be computed and verified in $\mathcal O(1.00000001^{n})$ with $n$ growing slowly with the input.
I know that there are hard problems in P(ex AKS principality testing) due to the time hierarchy theorem, wouldn't that also imply the existence of "easy" problems in EXPTIME?