I much prefer the concept of unbounded or indefinite for the physical concept.

So do I !!!

I’m not expert enough where mathematical infinities are concerned too be able to make any profound comments about what any of the more complex forms might be like, but for a true physical infinity, I think you have not gone far enough. If you apply Cantor's "infinity of infinities" to the Universe you would, as you say, have infinite numbers of everything, and everything that happens, happening an infinite number of times. However, this would be just a mathematical infinity. A mathematical ‘reality’ means only logical self-consistency and this does not require physical existence to validate it.

One often meets statements like: “In infinity everything that can happen will happen, an infinite number of times.” This statement contains at least two elements that are at variance with the concept of physical infinity: “…will happen”, because there can be no past or future in infinity; and “…number of times”, because there can be no numbers, or differentiated times in infinity. I said “at least two” because, strictly, “happen” has no place in infinity because it implies action, and action involves change.

How, then, can we re-word the original statement so as to bring it into line with the concept of physical infinity? It would have to be something like: “Every thing that can be, is”.