  We study the evolution of the maximum energy $E_\max(t)$ reached between time
$0$ and time $t$ in the dynamics of simple models with glassy energy
landscapes, in instant quenches from infinite temperature to a target
temperature $T$. Through a detailed description of the activated dynamics, we
are able to describe the evolution of $E_\max(t)$ from short times, through the
aging regime, until after equilibrium is reached, thus providing a detailed
description of the long-time dynamics. Finally, we compare our findings with
numerical simulations of the $p$-spin glass and show how the maximum energy
record can be used to identify the threshold energy in this model.
