  Using highly-accurate $N$-body simulations, we explore the evolution of
multiple massive black holes (hereafter, MBHs) in a primordial galaxy that is
composed of stars and MBHs. The evolution is pursed with a fourth-order Hermite
scheme, where not only three-body interaction of MBHs but also dynamical
friction by stars are incorporated. Initially, ten MBHs with equal mass of
$10^7M_\odot$ are set in a host galaxy with $10^{11}M_\odot$. It is found that
4 - 6 MBHs merge successively within 1 Gyr, emitting gravitational wave
radiation. The key process for the successive merger of MBHs is the dynamical
friction by field stars, which enhances three-body interactions of MBHs when
they enter the central regions of the galaxy. The heaviest MBH always composes
a close binary at the galactic center, which shrinks owing to the angular
momentum transfer by the third MBH and eventually merges. The angular momentum
transfer by the third MBH is due to the sling-shot mechanism. We find that the
secular Kozai mechanism does not work for a binary to merge if we include the
relativistic pericenter shift. The simulations show that a multiple MBH system
can produce a heavier MBH at the galactic center purely through $N$-body
process. This merger path can be of great significance for the growth of MBHs
in a primordial galaxy. The merger of multiple MBHs may be a potential source
of gravitational waves for the Laser Interferometer Space Antenna (LISA) and
pulsar timing.
