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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.