So far as I know, no one has proved the irrationality of Euler Mascheroni constant. There are discussions about the difficulty of proving the irrationality of this constant.
Since we cannot prove that this constant is irrational, is it not theoretically possible that this number is actually rational? Perhaps it can be written as the form $p/q$ where $q$ is a huge integer (far beyond the capacity of all current supercomputers). If so, it sounds to me like a prank on mathematicians from God.
Intuitively I also believe that this constant should be irrational; but isn't it also (perhaps extremely remotely) possible that it is rational? If so, all current effort in proving its irrationality is in the wrong direction.
Edit: Being reminded by a comment, I am basically asking "if it is rational, how does it affect mathematics"? For example, are there any theories based on the irrationality of this constant that need to be overthrown?