<p>The classical Gregory coefficients are also known as the (reciprocal) logarithmic numbers, the Cauchy numbers of the first kind or the Bernoulli numbers of the second kind. In this paper, we define Gregory coefficients of arbitrary order via the reciprocal of high powers of the natural logarithm and examine their many elegant properties analogous to those of the classical Gregory coefficients. In particular, we obtain several identities involving infinite series with higher-order Gregory coefficients and Euler’s (also known as Euler–Mascheroni’s) constant.</p>