  The behavior of the maximal displacement of a supercritical branching random
walk has been a subject of intense studies for a long time. But only recently
the case of time-inhomogeneous branching has gained focus. The contribution of
this paper is to analyze a time-inhomogeneous model with two levels of
randomness. In the first step a sequence of branching laws is sampled
independently according to a distribution on the set of point measures' laws.
Conditionally on the realization of this sequence (called environment) we
define a branching random walk and find the asymptotic behavior of its maximal
particle. It is of the form $V_n -\varphi \log n + o_\mathbf{P}(\log n)$, where
$V_n$ is a function of the environment that behaves as a random walk and
$\varphi>0$ is a deterministic constant, which turns out to be bigger than the
usual logarithmic correction of the homogeneous branching random walk.
