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.
It’s easy to use a uniform prior in a suitably huge interval - say from minus one gazillion to plus one gazillion - and you can look at the limit when the endpoints go to minus/plus infinity if you really have doubts about whether the interval was huge enough for your problem to fit comfortably within it.
If you don’t think that this is possible that says more about you than about the shortcomings of Bayesian statistics.
As I said, you can define the improper uniform prior solution as the limit of a sequence of solutions corresponding to a sequence of increasingly wider intervals with endpoints that go to +/-infinity.
(And as I said, you can start with a suitably huge region. Say that you want to determine the position of something and use a uniform prior that extends to a distance of 10^27m - a perfectly bounded prior from a mathematical point of view that covers the whole observable universe. If you observe something outside it, it’s not with the prior that you have a problem.)
