  The nonlinear memory effect is a fascinating prediction of general relativity
(GR), in which oscillatory gravitational-wave (GW) signals are generically
accompanied by a monotonically-increasing strain which persists in the detector
long after the signal has passed. This effect presents a unique opportunity to
test GR in the dynamical and nonlinear regime. In this article we calculate the
nonlinear memory signal associated with GW bursts from cusps and kinks on
cosmic string loops, which are an important target for current and future GW
observatories. We obtain analytical waveforms for the GW memory from cusps and
kinks, and use these to calculate the "memory of the memory" and other
higher-order memory effects. These are among the first memory observables
computed for a cosmological source of GWs, with previous literature having
focused almost entirely on astrophysical sources. Surprisingly, we find that
the cusp GW signal diverges for sufficiently large loops, and argue that the
most plausible explanation for this divergence is a breakdown in the weak-field
treatment of GW emission from the cusp. This shows that previously-neglected
strong gravity effects must play an important role near cusps, although the
exact mechanism by which they cure the divergence is not currently understood.
We show that one possible resolution is for these cusps to collapse to form
primordial black holes (PBHs); the kink memory signal does not diverge, in
agreement with the fact that kinks are not predicted to form PBHs. Finally, we
investigate the prospects for detecting memory from cusps and kinks with GW
observatories. We find that in the scenario where the cusp memory divergence is
cured by PBH formation, the memory signal is strongly suppressed and is not
likely to be detected. However, alternative resolutions of the cusp divergence
may in principle lead to much more favourable observational prospects.
