Berdan primed cases are reloadable, but it is not as simple as Boxer primed cases and not generally done in the US. Berdan primers are not easily found in the US and i understand there are more different sizes than the standard Boxer primers.
At one time CCI made some aluminum cased ammunition (they may still) that was primed with a primer not the same size as the standard small or large primers. The idea was to prevent the case from being reloaded. I suppose there was concern over the safety of reloading the aluminum.
Remington made an electronically fired cartridge a few years ago. Not sure if the cartridges are still available, but it did not set the world on fire.
i suppose there have been other cartridges made with non-standard primers but don't remember any off hand. At least in recent times, it would probably be the death of a cartridge as the reloading community would not support it.