  Quasi-topological terms in gravity can be viewed as those that give no
contribution to the equations of motion for a special subclass of metric
ans\"atze. They therefore play no r\^ole in constructing these solutions, but
can affect the general perturbations. We consider Einstein gravity extended
with Ricci tensor polynomial invariants, which admits Einstein metrics with
appropriate effective cosmological constants as its vacuum solutions. We
construct three types of quasi-topological gravities. The first type is for the
most general static metrics with spherical, toroidal or hyperbolic isometries.
The second type is for the special static metrics where $g_{tt} g_{rr}$ is
constant. The third type is the linearized quasi-topological gravities on the
Einstein metrics. We construct and classify results that are either dependent
on or independent of dimensions, up to the tenth order. We then consider a
subset of these three types and obtain Lovelock-like quasi-topological
gravities, that are independent of the dimensions. The linearized gravities on
Einstein metrics on all dimensions are simply Einstein and hence ghost free.
The theories become quasi-topological on static metrics in one specific
dimension, but non-trivial in others. We also focus on the quasi-topological
Ricci cubic invariant in four dimensions as a specific example to study its
effect on holography, including shear viscosity, thermoelectric DC
conductivities and butterfly velocity. In particular, we find that the
holographic diffusivity bounds can be violated by the quasi-topological terms,
which can induce an extra massive mode that yields a butterfly velocity unbound
above.
