There are a lot of random variables in physics that are, in principle, continuous and unbounded (canonical examples: time, position along each direction). In practice, it is my understanding that the number of outcomes of every possible experiment will be a finite integer. That is, the data is always binned, even if only implicitly, at some minimum scale and finite in scope.
Reading the length of something off of a ruler, for example, will always produce a number that is rounded to some nearest rational number, whether that number is a line on the ruler or interpolated from the scale by eye. It will also always only produces either a finite length answer or a lower limit, assuming the experiment has to be carried out in finite time (i.e. using a single ruler you can only translate it a finite number of times to measure something longer than it).
In principle, though, the length of something being measured by a ruler can assume a continuum of values, but this question isn't about the theoretical principle, it's about experiments that are realizable in finite time using finite resources. Is there any example of a real experiment that has an actual infinite number of outcomes in any way? If not, is there a good reason to think that we will never be able to produce such an experiment/measurement?