A sample of 542 eclipsing binaries (EBs) with periods shorter than 2 days was selected from the Microlensing Observations in Astrophysics (MOA) EB catalogue (Li et al.) for eclipse-time variation analysis. For this sample, we were able to obtain a time series from MOA-II spanning 9.5 yr. We discovered 91 EBs, out of the 542 EBs, with detected light-travel-time effect signals suggesting the presence of tertiary companions with orbiting periods from 250 d–28 yr. The frequency of EBs with tertiary companions in our sample increases as the period decreases and reaches a value of 0.65 for contact binaries with periods shorter than 0.3 d. If only those contact binaries with periods <0.26 d are considered, the frequency even goes to unity. Our results suggest that contact binaries with periods close to the 0.22-d contact binary limit are commonly accompanied by relatively close tertiary companions. [ABSTRACT FROM AUTHOR]