  We propose a new model of Early Dark Energy (EDE) as a solution to the Hubble
tension in cosmology, the apparent discrepancy between local measurements of
the Hubble constant $H_0\simeq 74$ km s$^{-1}$ Mpc$^{-1}$ and $H_0\simeq 67$ 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 $\sim 10\%$ 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 (meV 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.
