We propose a new model of early dark energy (EDE) as a possible solution to the Hubble tension in cosmology, the apparent discrepancy between local measurements of the Hubble constant H-0 similar or equal to 74 km s(-1) Mpc(-1) and H-0 similar or equal to 67 km s(-1) Mpc(-1) inferred from the cosmic microwave background (CMB). In chain EDE, the universe undergoes a series of first order phase transitions, starting at a high energy vacuum in a potential, and tunneling down through a chain of every lower energy metastable minima. As in all EDE models, the contribution of the vacuum energy to the total energy density of the universe is initially negligible, but reaches similar to 10% around matter-radiation equality, before cosmological data require it to redshift away quickly-at least as fast as radiation. We indeed obtain this required behavior with a series of N tunneling events, and show that for N > 600 the phase transitions are rapid enough to allow fast percolation and thereby avoid large scale anisotropies in the CMB. We construct a specific example of chain EDE featuring a scalar field in a quasiperiodic potential (a tilted cosine), which is ubiquitous in axion physics and, therefore, carries strong theoretical motivation. Interestingly, the energy difference between vacua can be roughly the size of today's dark energy (milli-electron-volt scale). Therefore, the end result of chain EDE could provide a natural explanation of dark energy, if the tunneling becomes extremely slow in the final step before the field reaches zero (or negative) energy. We discuss a simple mechanism which can stop the scalar field in the desired minimum. Thus chain EDE offers the exciting prospect to explain EDE and dark energy by the same scalar field.