Scaling of the exact function for the number of intramolecular nonbonded contacts in a single maximally compact linear homopolymer on hypercubic lattices is determined as a function of number N of monomers and dimension d. A representative maximally compact structure is designed and an exact recursive expression for the maximum number m(max) of contacts is derived from that design. The equivalent nonrecursive expression yields the asymptotic scaling of m(max) as (d-1)N-dN(Delta)+1, with Delta=(d-1)/d. Implications in polymer and protein studies are discussed.