We report results on the reactions p-\-p -> K+K+nir (n= 1, 2, • • •) at 2.7 GeV/c. The cross section for this process is estimated to be 4.1 ±0.6 mb. Comparison of 2.7-GeV/c partial cross sections with data at 3.0 and 3.7 GeV/c indicates that these partial cross sections either are relatively constant or slowly decrease with energy. K*(890) production is observed in all the four-, five-, and six-body final states. The production of p° and oi° is also observed. The KK mass spectra show enhancements near threshold as has been seen in other p-p experiments. The present data cannot distinguish between a resonance or S-wave scattering-length interpretation of these enhancements. No significant evidence for resonances decaying into Kinr or KKw was observed.