  Persistent homology captures the appearances and disappearances of
topological features such as loops and holes when growing disks centered at a
Poisson point process. We study extreme values for the life times of features
dying in bounded components and with birth resp. death time bounded away from
the threshold for continuum percolation. First, we describe the scaling of the
minimal life times for general feature dimensions, and of the maximal life
times for holes in the \v{C}ech complex. Then, we proceed to a more refined
analysis and establish Poisson approximation for large life times of holes and
for small life times of loops. Finally, we also study the scaling of minimal
life times in the Vietoris-Rips setting and point to a surprising difference to
the \v{C}ech complex.
