I will introduce Chain inflation as an alternative to slow roll inflation in which the universe undergoes a series of transitions between different vacua. The role of the inflaton can be played by an axion which tunnels from minimum to minimum in a quasiperiodic potential. I will determine the scalar power spectrum of chain inflation and show that it is fully consistent with a ΛCDM cosmology. Then I will turn to the Hubble tension, the apparent discrepancy between local measurements of the Hubble constant H0=74 km/s/Mpc and H0=67 km/s/Mpc inferred from the CMB. I will propose Chain Early Dark Energy - the low-energy analog of chain inflation - as a solution of the H0-tension and point to a connection to today's dark energy.