  Ultralight bosons can be abundantly produced through superradiance process by
a spinning black hole and form a bound state with hydrogen-like spectrum. We
show that such a "gravitational atom" typically possesses anomalously large
mass quadrupole and leads to significant orbital precession when it forms an
eccentric binary with a second compact object. Dynamically formed black hole
binaries or pulsar-black hole binaries are typically eccentric during their
early inspirals. We show that the large orbital precession can generate
distinct and observable signature in their gravitational wave or pulsar timing
signals.
