  We put forward novel extensions of Starobinsky inflation, involving a class
of 'geometric' higher-curvature corrections that yield second-order
Friedmann-Lema\^itre equations and second-order-in-time linearized equations
around cosmological backgrounds. We determine the range of models within this
class that admit an extended phase of slow roll inflation as an attractor. By
embedding these theories in anti-de Sitter space, we derive holographic
'unitarity' bounds on the two dominant higher-order curvature corrections.
Finally we compute the leading corrections to the spectral properties of scalar
and tensor primordial perturbations, including the modified consistency
relation $r=-8n_{T}$. Remarkably, the range of models singled out by holography
nearly coincides with the current observational bounds on the scalar spectral
tilt. Our results indicate that future observations have the potential to
discriminate between different higher-curvature corrections considered here.
