Euler's constant γ is one of the mathematical constants with the most different analytic representations, probably on par with π. Yet, none of these representations proves that γ is a value of an E-function, a G-function or an M-function at an algebraic point. In fact, it is plausible that no such representation of γ exists with these three arithmetic special functions, and thus the arithmetic nature of γ might not be determined by the powerful Diophantine theorems of Siegel-Shidlovsky, Chudnovsky and Nishioka. Nonetheless, we explain here why certain of these representations show that γ is not far from being a special value of both E-functions and M-functions, while a similar connection to G-functions is still elusive. We also present a new family of series summing to γ, which generalize an identity of Vacca.