We introduce the idea of an "equi-g(r) sequence." This consists of a series of equilibrium many-body systems which have different number densities rho but share, at a given temperature, the same form of pair correlation function, termed "target g(r)." Each system is defined by a pair potential indexed by rho as in u(rho)(r). It is shown that for such a sequence a terminal density rho(star) exists, beyond which no physically realizable system can be found. As an illustration we derive explicit values of rho(star) for target g(r) that is based on a square-well potential in the limit rho-->0. Possible application of this terminal phenomenon to the investigation into limiting amorphous packing structures of hard spheres is proposed. Virial expansions of u(rho)(r) and pressure are carried out and compared with the corresponding expressions for imperfect gas. The behaviors of u(rho)(r) and pressure close to rho=rho(star) are examined as well, and associated exponents extracted when they exist. The distinction between equi-g(r) sequence and the related, recently introduced concept of "iso-g((2)) process" is briefly discussed.