  This is the first of a three paper series in which we present a comprehensive
study of the extreme value theory of the scale-inhomogeneous discrete Gaussian
free field. This model was introduced by Arguin and Ouimet who computed the
first order of the maximum. In this first paper we establish tail estimates for
the maximum value, which allow to deduce the log-correction to the order of the
maximum and tightness of the centred maximum. Our proofs are based on the
second moment method and Gaussian comparison techniques.
