  In Chain Inflation the universe tunnels along a series of false vacua of
ever-decreasing energy. The main goal of this paper is to embed Chain Inflation
in high energy fundamental physics. We begin by illustrating a simple effective
formalism for calculating Cosmic Microwave Background (CMB) observables in
Chain Inflation. Density perturbations seeding the anisotropies emerge from the
probabilistic nature of tunneling (rather than from quantum fluctuations of the
inflation). To obtain the correct normalization of the scalar power spectrum
and the scalar spectral index, we find an upper limit on the scale of inflation
at horizon crossing of CMB scales, $V_*^{1/4}< 10^{12}$~GeV. We then provide an
explicit realization of chain inflation, in which the inflaton is identified
with an axion in supergravity. The axion enjoys a perturbative shift symmetry
which is broken to a discrete remnant by instantons. The model, which we dub
`natural chain inflation' satisfies all cosmological constraints and can be
embedded into a standard $\Lambda$CDM cosmology. Our work provides a major step
towards the ultraviolet completion of chain inflation in string theory.
