I will discuss the directed polymer in Gaussian environment in dimension $2$, with subcritical scaling that leads to weak disorder. It is known since the work of Caravenna, Sun and Zygouras that the logarithm of the partition function, properly scaled, converges to a logarithmically correlated field. The study of extremes of this field leads to some surprises and helps in elucidating the nature of the field, which relates to branching random walks with (slowly) varying variance profile. (Work in progress with Clement Cosco and Shuta Nakajima)