I think you misunderstand Hilbert's Infinite Hotel Paradox. The point is, in mathematics a "paradox" is just a fancy way of describing a contradiction. When you reach a contradiction, logic demands that you back-track through your reasoning to find an error (incorrect assumption or deduction). In Hilbert's case it was the assumption that you can do normal arithmetic with the mathematical concept of infinity. This led to the development of transfinite numbers where things work rather differently etc. But there is no real problem here, and it certainly does not demostrate that infinity doesn't exist in some sense.
However, to adress you question directly: I'm not sure. (And I'm a mathematician.) I can think of 2 good candidates for examples of an "infinity" in reality: either time might be of infinite "length" (as argued elsewhere); or the infinite divisibility of space there could provide an example of an infinite collection. Not sure if, in reality, space is made up of infinitely many infinitessimal "bits" or if there's a minimal size to things. In maths, a line does have an infinite number of points along it (even in a line of finite length - see Zeno's Paradox). I suspect there is some minimum size to things, but completely without foundation of course.
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