We present the first fully relativistic longterm numerical evolutions of three equal-mass black holes in a system consisting of a third black hole in a close orbit about a black-hole binary. We find that these close-three-black-hole systems have very different merger dynamics from black-hole binaries. In particular, we see complex trajectories, a redistribution of energy that can impart substantial kicks to one of the holes, distinctive waveforms, and suppression of the emitted gravitational radiation. We evolve two such configurations and find very different behaviors. In one configuration the binary is quickly disrupted and the individual holes follow complicated trajectories and merge with the third hole in rapid succession, while in the other, the binary completes a half-orbit before the initial merger of one of the members with the third black hole, and the resulting two-black-hole system forms a highly elliptical, well separated binary that shows no significant inspiral for (at least) the first t ∼ 1000M of evolution. Introduction: The recent dramatic breakthroughs in the numerical techniques to evolve black-hole-binary spacetimes [1, 2, 3] has led to rapid advancements in our understanding of black-hole physics. Notable among these advancements are developments in mathematical relativity, including systems of PDEs and gauge choices [4,5,6], the exploration of the cosmic censorship [7,8,9,10,11], and the application of isolated horizon formulae [8,9,12,13,14,15]. These breakthroughs have also influenced the development of data analysis techniques through the matching of post-Newtonian to fully-numerical waveforms [16,17,18]. Similarly, the recent discovery of very large merger recoil kicks [19,20,21,22,23,24,25,26] has had a great impact in the astrophysical community, with several groups now seeking for observational traces of such high speed holes as the byproduct of galaxy collisions [27,28]. In this letter, we continue our quest to discover new astrophysical consequences of black-hole interactions by simulating close encounters of three black holes to see the different behaviors introduced by the finite size of the holes, their nonlinear interactions, and the radiation of gravitational waves, as described by General Relativity. We find that the three-body relativistic problem shows far richer dynamics than the two-body problem, akin to the rich three-body dynamics in Newtonian gravity, but with added complexity due to mergers.