In nature, it is believed that astrophysical black holes will be nearly neutral and highly spinning. The first condition is due to the fact that, if the black hole were charged, it would preferentially attract opposing charges. The second follows from considerations of matter accreting onto the black hole. By conservation of angular momentum, it is expected that it might be common for black hole spins to approach their theoretical maximal value. In this regime, the non-linearity of Einstein's equations comes into play, and effects such as frame dragging become strong. This becomes particularly interesting in the case of a black hole binary, in which emission of gravitational waves and black hole kicks become important. To date, the simulations with the largest black hole spins fail to probe the top ~35% of the maximal rotational energy. Thus, there is a substantial unexplored energy range that is of real astrophysical and theoretical interest. We use numerical methods to solve Einstein's equations, which enable us to describe the motions of black holes under the influence of gravitation.