  We explore the asymptotic future evolution of holographic dark energy (HDE)
models, in which the density of the dark energy is a function of a cutoff scale
$L$. We develop a general methodology to determine which models correspond to
future big rip, little rip, and pseudo-rip (de Sitter) evolution, and we apply
this methodology to a variety of well-studied HDE models. None of these HDE
models display little rip evolution, and we are able to show, under very
general assumptions, that HDE models with a Granda-Oliveros cutoff almost never
evolve toward a future little rip. We extend these results to HDE models with
nonstandard Friedman equations and show that a similar conclusion applies:
little rip evolution is a very special case that is almost never realized in
such models.
