In this Letter we calculate the fraction of highly eccentric binary black hole (BBH) mergers resulting from binary-single interactions. Using an N-body code that includes post-Newtonian correction terms, we show that ≳1% of all BBH mergers resulting from this channel will have an eccentricity e > 0.1 when coming into the LIGO frequency band. As the majority of BBH mergers forming in globular clusters are assembled through three-body encounters, we suggest that such interactions are likely to dominate the population of high-eccentricity BBH mergers detectable by LIGO. The relative frequency of highly eccentric events could eventually help to constrain the astrophysical origin of BBH mergers.