Intermediate problems between P and NP are quite famous, and are sometimes considered as complexity classes by themselves.
Do you know of any problem that is known to be PSPACE-hard and in EXPTIME, and resisting all efforts to be proved complete for one of these classes ?
Lifted (succinct) versions of problems between LOGSPACE and PTIME are accepted, but even more interesting would be problems that are not of this form.
Here are some I found, this area of complexity theory seems to be the realm of games:
- Phutball: https://arxiv.org/abs/0804.1777
- Go (with Chinese rules): D. Lichtenstein and M. Sipser, Go is polynomial-space hard