  Recent analyses of cosmic microwave background surveys have revealed hints
that there may be a non-trivial running of the running of the spectral index.
If future experiments were to confirm these hints, it would prove a powerful
discriminator of inflationary models, ruling out simple single field models. We
discuss how isocurvature perturbations in multi-field models can be invoked to
generate large runnings in a non-standard hierarchy, and find that a minimal
model capable of practically realising this would be a two-field model with a
non-canonical kinetic structure. We also consider alternative scenarios such as
variable speed of light models and canonical quantum gravity effects and their
implications for runnings of the spectral index.
