Chain inflation is an alternative to slow roll inflation in which the Universe undergoes a series of transitions between different vacua. The density perturbations (studied in this paper) are seeded by the probabilistic nature of tunneling rather than quantum fluctuations of the inflaton. We find the scalar power spectrum of chain inflation and show that it is fully consistent with a Lambda CDM cosmology. In agreement with some of the previous literature (and disagreement with others), we show that 10(4) phase transitions per e-fold are required in order to agree with the amplitude of cosmic microwave background anisotropies within the observed range of scales. Interestingly, the amplitude of perturbations constrains chain inflation to a regime of highly unstable de Sitter spaces, which may be favorable from a quantum gravity perspective since the swampland conjecture on trans-Planckian censorship is automatically satisfied. We provide new analytic estimates for the bounce action and the tunneling rate in periodic potentials which replace the thin-wall approximation in the regime of fast tunneling. Finally, we study model implications and derive an upper limit of similar to 10(10) GeV on the axion decay constant in viable chain inflation with axions.