The only police officers that I know of that do not have police powers anywhere in the state would be any University Police Officer. They have the exact same training, but due to the fact that it's a University police department, that is not non-profit, the state does not allow it. The way around this is "jurisdictional agreements" with surrounding agencies, and in some cases, University police officers being "sworn in" as reserve deputy sheriff's for the purposes of having powers to assist those agencies. I believe this was up for debate a few years ago, that may have changed and UPD officers are now considered like the rest.
The newly formed gaming commission officers may not have full powers either, but I'm not sure.
I'm a full-time special deputy for a sheriff's department. By policy, my powers stop at the end of county property. I do have full arrest and police powers though, because I'm a deputy town marshal for another agency. Reserves, Town Marshals, Reserve Marshals, Special Deputies, IDNR COs, State Troopers, Excise officers, City Officers, Railroad Police, and State Fire Marshals all have full police/arrest powers AFAIK...