The nonadditive dispersion contribution to the binding energy of three one-dimensional (1D) wires is investigated using wires modeled by (i) chains of hydrogen atoms and (ii) homogeneous electron gases. We demonstrate that the nonadditive dispersion contribution to the binding energy is significantly enhanced compared with that expected from Axilrod-Teller-Muto–type triple-dipole summations and follows a different power-law decay with separation. The triwire nonadditive dispersion for 1D electron gases scales according to the power law d^(-beta), where d is the wire separation, with exponents beta(r_s) smaller than 3 and slightly increasing with r_s from 2.4 at r_s=1 to 2.9 at r_s=10, where r_s is the density parameter of the 1D electron gas. This is in good agreement with the exponent beta=3 suggested by the leading-order charge-flow contribution to the triwire nonadditivity, and is a significantly slower decay than the ~d^(-7) behavior that would be expected from triple-dipole summations.