  We discuss the possible existence of the fully-heavy tetraquarks. We
calculate the ground-state energy of the $bb \bar b \bar b$ bound state, where
$b$ stands for the bottom quark, in a nonrelativistic effective field theory
framework with one-gluon-exchange (OGE) color Coulomb interaction, and in a
relativized diquark model characterized by OGE plus a confining potential. Our
analysis advocates the existence of uni-flavor heavy four-quark bound states.
The ground state $bb\bar b\bar b$ tetraquark mass is predicted to be
$(18.72\pm0.02)$~GeV. Mass inequality relations among the lowest
$QQ\bar{Q}\bar{Q}$ state, where $Q\in \{c, b\}$, and the corresponding heavy
quarkonia are presented, which give the upper limit on the mass of ground state
$QQ\bar{Q}\bar{Q}$. The possible decays of the lowest $bb\bar{b}\bar{b}$ are
highlighted, which might provide useful references in the search for them in
ongoing LHC experiments, and its width is estimated to be a few tens of MeV.
