  We derive an analytic expression of the effective potential at finite
temperature (T) and chemical potential (mu) in the strong-coupling lattice QCD
for color SU(3) including next-to-next-to-leading order (NNLO) effects in the
strong coupling expansion. NNLO effective action terms are systematically
evaluated in the leading order of the large dimensional (1/d) expansion, and
are found to come from some types of connected two plaquette configurations. We
apply the extended Hubbard-Stratonovich transformation and a gluonic dressed
fermion technique to the effective action, and obtain the effective potential
as a function of T, mu, and two order parameters; chiral condensate and a
vector potential field. The next-to-leading order (NLO) and NNLO effects result
in modifications of the wave function renormalization factor, quark mass and
chemical potential. We find that T_{c,mu=0} and mu_{c,T=0} are similar to the
NLO results, whereas the position of the critical point is sensitive to NNLO
corrections.
