  The $R^2$ inflation which is an extension of general relativity (GR) by
quadratic scalar curvature introduces a quasi-de Sitter expansion of the early
Universe governed by Ricci scalar being an eigenmode of d'Alembertian operator.
In this paper, we derive a most general theory of gravity admitting $R^2$
inflationary solution which turned out to be higher curvature non-local
extension of GR. We study in detail inflationary perturbations in this theory
and analyse the structure of form-factors that leads to a massive scalar
(scalaron) and massless tensor degrees of freedom. We argue that the theory
contains only finite number of free parameters which can be fixed by
cosmological observations. We derive predictions of our generalized non-local
$R^2$-like inflation and obtain the scalar spectral index $n_s\approx
1-\frac{2}{N}$ and any value of the tensor-to-scalar ratio $r<0.036$. In this
theory, tensor spectral index can be either positive or negative $n_t\lessgtr
0$ and the well-known consistency relation $r = -8n_t$ is violated in a
non-trivial way. We also compute running of the tensor spectral index and
discuss observational implications to distinguish this model from several
classes of scalar field models of inflation. These predictions allow us to
probe the nature of quantum gravity in the scope of future CMB and
gravitational wave observations. Finally we comment on how the features of
generalized non-local $R^2$-like inflation cannot be captured by established
notions of the so-called effective field theory of single field inflation and
how we must redefine the way we pursue inflationary cosmology.
