Nice. In order to assess if infinities are physically possible, we should first ask if (physical) finities are actually possible. The question assumes such fact, but that's debatable.
Finite numbers are a systemic mental construct: they are defined by ideal boundaries. For example, we take a continuous line, and split it in similar parts. That is our representation of the physical nature. If we find boundaries, we can count things, or formally systems. Although systems are just mental constructs.
For example, if there are few clouds in the sky, you can count them, and you can say that there is not an infinite number of clouds. But what happens if the sky is completely covered? Would you say that there's an infinite number of clouds? Or would you say there's just one cloud? Or that there's zero clouds?
Time and space are similar entities: sometimes we perceive them as discrete chunks, other times as continuous stuff. So we have learned to enumerate them. For example, a minute is a discretization of time. The equivalence with clouds in a rainy day would be to draw a grid on the sky and count clouds if cells are occupied.
There's an idea I enjoy exploring: physical nature would be like a number, but without the decimal point. What is the meaning of that? What would be a number without integer and fractional parts? But in fact, the problem comes from the other side: why have we chosen to make integers out of nature? Why did we created the decimal point? What is the point of numbering things? That's because our mind needs to define borders, limits, boundaries, frontiers in order to interact with nature. A cloud or a rainbow exist as an integer unit... depending on my subjective physical location, my perception, my memory, the scale of my existence. The same happens with a river. Or a tree. Or a rock. It seems that the borders of a rock are much more defined than the borders of a rainbow, but it's just a matter of scales. Things don't exist physically.
So, everything happens in our subjective perceptions. So, perhaps the real question is... are finities actually possible? And my personal answer is no. We have discretized matter in our minfs, but physically, everything is just energy, has no boundaries. In consequence, finities are not possible. Ergo, infinities cannot be physically possible. It is enough to count the clouds in the sky. Perhaps you and me can agree on the number, but does that physically mean something?
Update: two metallic objects can be put together and they keep being two objects. But the only reason they keep being two separate entities is because there's air between the surfaces. If two objects are joined in space, they become one; "there is no way for the atoms to ‘know’ that they are in different pieces of [metal]" (Richard Feynman). So, the apparent "number" of parts is just a subjective appreciation. Finite entities are apparent to our perception, but that has no physical meaning.
Or perhaps you mean that if an eternally-living person were able to count the number if clouds in all stars (after defining a precise taxonomy of clouds), it will never finish. That is out of our current knowledge. Perhaps she will only be able to count a finite number of clouds... forever.