  Recently, A. I. Aptekarev and his collaborators found a sequence of rational
approximations to Euler's constant $\gamma$ defined by a third-order
homogeneous linear recurrence. In this paper, we give a new interpretation of
Aptekarev's approximations in terms of Meijer $G$-functions and
hypergeometric-type series. This approach allows us to describe a very general
construction giving linear forms in 1 and $\gamma$ with rational coefficients.
Using this construction we find new rational approximations to $\gamma$
generated by a second-order inhomogeneous linear recurrence with polynomial
coefficients. This leads to a continued fraction (though not a simple continued
fraction) for Euler's constant. It seems to be the first non-trivial continued
fraction expansion convergent to Euler's constant sub-exponentially, the
elements of which can be expressed as a general pattern. It is interesting to
note that the same homogeneous recurrence generates a continued fraction for
the Euler-Gompertz constant found by Stieltjes in 1895.
