We consider graviton scattering in maximal supergravity on Anti-de Sitter space (AdS) in d+1 dimensions for d=3,4,and 6 with no extra compact spacetime factor. Holography suggests that this theory is dual to an exotic maximally supersymmetric conformal field theory (CFT) in d dimensions whose only light single trace operator is the stress tensor. This contrasts with more standard cases like Type IIB string theory on AdS_5x S^5 dual to N=4 Super-Yang-Mills, where the CFT has light single trace operators for each Kaluza-Klein mode on S^5. We compute the 1-loop correction to the pure AdS_{d+1} theory in a small Planck length expansion, which is dual to the large central charge expansion in the CFT. We find that this correction saturates the most general non-perturbative conformal bootstrap bounds on this correlator in the large central charge regime for d=3,4,6. After imposing theory-specific constraints from localization in d=3,4, the bootstrap constraints strengthen and are then saturated by the string/M-theory dual CFT data.