Sometimes people fail at asking a question by being too broad, unclear like here. So I'll take a stab at what I believe to be the same question, but more concise and clearly stated:
Does infinity exist in either terms of structure, parameters, or measurement in any physical systems?
Can infinity have any real connection with reality?
Or is infinity purely a mathematical concept just used by physicists as a convenient way to describe the very large, an approximation?
I have heard that if you model a physical system (recently Brian Greene posted a video on YouTube regarding infinity), and you run into infinity as a solution, then you have either made an error in your calculations or your model is wrong.