Very recently, a fundamental observable has been introduced and analyzed to quantify the exploration of random walks: the time τ_$k$ required for a random walk to find a site that it never visited previously, when the walk has already visited $k$ distinct sites. Here, we tackle the natural issue of the statistics of Mn, the longest duration out of τ₀,…,τ_n-1. This problem belongs to the active field of extreme value statistics, with the difficulty that the random variables τ_$k$ are both correlated and non-identically distributed. Beyond this fundamental aspect, we show that the asymptotic determination of the statistics of M_n finds explicit applications in foraging theory and allows us to solve the open d-dimensional starving random walk problem, in which each site of a lattice initially contains one food unit, consumed upon visit by the random walker, which can travel S steps without food before starving. Processes of diverse nature, including regular diffusion, anomalous diffusion, and diffusion in disordered media and fractals, share common properties within the same universality classes.