  The $B_c$ meson pair, including pairs of pseudoscalar states and vector
states, productions in proton-proton collisions are investigated at the
next-to-leading order (NLO) accuracy in the nonrelativistic quantum
chromodynamics factorization formalism. The corresponding cross sections at the
Large Hadron Collider (LHC) with $\sqrt{s}=14\; \text{TeV}$ are evaluated.
Numerical results indicate that the NLO corrections are substantial, and even
dominate over the leading order contributions. Considering the predicted cross
sections are sizable, the $B_c$-pair production is expected to be observable at
the High-Luminosity LHC experiment.
