Can you give an example of a problem where there is no possible prior information, and states all the way from negative to positive infinity are equally plausib

Can you give an example of a problem where there is no possible prior information, and states all the way from negative to positive infinity are equally plausible? I don't think the laws of physics (not to mention, the practical limits of computation) allow for real world inference problems where there are no bounds on a prior of any kind. Failing that, I don't think there are any human usable data sources that could report observations over an infinite interval.