  We introduce and study finite analogues of Euler's constant in the same
setting as finite multiple zeta values. We define a couple of candidate values
from the perspectives of a ``regularized value of $\zeta(1)$'' and of
Mascheroni's and Kluyver--N\"orlund's series expressions of Euler's constant
using Gregory coefficients. Moreover, we reveal that the differences between
them always lie in the $\mathbb{Q}$-vector space spanned by 1 and values of a
finite analogue of logarithm at positive integers.
