I can't think of a single thing that could be infinite.
Because the universe is expanding, isn't it? But there is an ever-changing barrier, so why could there be anything infinite, both hypothetically and realistically?
This is a philosophical question so here is a philosophical answer.
The scientific method in based on repeated observations and experiment. The whole science is just a collectivist instrument of acquiring knowledge. Being an instrument, it has its own limitations. Among them are:
This means that there is always a limit on the largest value that can be scientifically measured.
So the conclusion is: science (that is, physics) cannot establish existence of infinite quantities. There is nothing physically infinite. As such, existence or non-existence of infinite quantities is outside of the scope of science.
No (understandable/explainable) physical quantity could be infinite. "Infinity" is is physically very vague. When we say something is "infinite", it almost means we're throwing our hands up in despair that we can't explain something, or that quantity doesn't make sense in some particular framework. The whole point of physical quantities (observables) is to let us characterize physical phenomena. And a quantity which is infinite is pretty much useless for that purpose.
For starters, there are different kinds of infinities. The correct way to think about the concept of infinity is as a limit as something increasing, but without an upper bound. So it can grow arbitrarily large -- aka "infinity". So even if something is growing arbitrarily large, we can ask how fast it's growing and as compared to what other physical quantity (or some parameters). We can then replace the naively infinite physical quantity with this new quantity that tells us how fast it is going to infinity.
For all our progress in physics, we're still making effective theories to model phenomena in some particular regime. So in some cases, it might well be that our effective theories don't make sense beyond some values of the parameters. So we'll have to "cut off" the validity of our theory in some regime of the parameters and claim that beyond that, some better theory must take over.
Ultimately, the guiding principle is that nature can never behave badly enough to make some physical quantity infinite. If you see a physical quantity becoming arbitrarily large (ie without bound), then you're doing something wrong -- either considering the wrong quantity, or using a framework beyond it's regime of validity.
How about space? I understand a standard assumption in cosmology is that the universe is spatially infinite in extent.
Of course whether it actually is - is an entirely separate question.
For how can you measure something to be infinite? It maybe that some parameter is inversely related to that something. And you could measure that value to be zero.
In Philosophy Aristotle stated that Infinities can never be actual but only potential.
Here is a list of actual infinities in physics, mentioning
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