Is infinity real? Depends on what you mean by real and infinity. The basis for this question is to ask whether there are any actual infinities. Not just potential ones as in many domains of mathematics, but quantities or durations that are infinite. This includes anything "infinitely small" that isn't actually nothing (as in ontological non-existence).
From a mathematical standpoint, the real numbers cannot be defined without infinities, which is to say that an actual real number as distinct from the infinite sets of rational numbers infinitesmally close. In other words, if infinity isn't real, then neither are the irrational reals like π and square_root(2).
None of this is much of a problem unless the mathematics changes significantly when you substitute rational number for real number to acknowledge the unreality of infinity.
We also can consider the question from the standpoint of eternity and timelessness.
Following Peirce, all inquiries are to be conducted as semiotic inquiries. To say that actual infinities don't exist, but that potential ones do is very easy to express in these terms. The real existence of infinity is in the position of object in the sign indicated by the english word infinity which is interpreted according to one of more discourses on infinity. It really is a sign object use by mathemeticians to talk about potential infinities. The discource about whether any actual infinities exist is a metaphisical one. It is surely an interesting metaphisical question because science may be able to test for it. No current experiments or even proposals address it directly ... yet.
The Wittgenstein quote takes us to the deeper question. Pragmatism will bring forward the way signs create meaning by the connection of concrete chunks that are integrated semi-sequentially. Final answers to any rational program withing pragmatism can only come at the end of a potentially infinite span. Timelessness is an idealization and projection of finite processes to infinite ones (and if I'm not being philosophically ignorant, this is really what second order logic is all about). If the Timeless Present is a better description of the reality of how an experiential world unfolds, then this is an example of a real infinity. Is it unreasonable to suggest that we can experience infinity in the eternal present? What would count as evidence for this being true?