  In this paper we study perturbatively an extension of the Stelle higher
derivative gravity involving an infinite number of derivative terms. We know
that the usual quadratic action is renormalizable but suffers of the unitarity
problem because of the presence of a ghost (state of negative norm) in the
theory. The new theory is instead ghost-free since the introduction of (in
general) two entire functions in the model with the property do not introduce
new poles in the propagator. The local high derivative theory is recovered
expanding the entire functions to the lowest order in the mass scale of the
theory. Any truncation of the entire functions gives rise to the unitarity
violation but if we keep all the infinite series we do not fall into these
troubles. The theory is renormalizable at one loop and finite from two loops
on. Since only a finite number of graphs are divergent then the theory is
super-renormalizable. We analyze the fractal properties of the theory at high
energy showing a reduction of the spacetime dimension at short scales. Black
hole spherical symmetric solutions are also studied omitting the high curvature
corrections in the equation of motions. The solutions are regular and the
classical singularity is replaced by a "de Sitter-like core" in r=0. Black
holes may show a "multi-horizon" structure depending on the value of the mass.
