Perhaps I'm in way over my head asking this; regardless, considering the abundance of known formulas and properties of $\gamma$, seemingly comparable in quantity to the likes of $\pi$ or $e$, why is the irrationality of such so hard to prove?
In other words, what exactly makes the surrounding mysteries of the constant out of reach of present-day mathematics? Or is the mystery a mystery in itself?
I note that user571036 asked a similar question, but specifically surrounding one method of proving it's irrationality (using it's relationship to the harmonic series).