Strong, long-range Coulomb interactions can lead to correlated motions of multiple charged particles, which can induce important many-body effects in semiconductors. The exciton states formed from correlated electron-hole pairs have been studied extensively, but basic properties of multiple-exciton correlations-such as coherence times, population lifetimes, binding energies and the number of particles that can be correlated-are largely unknown because they are not spectroscopically accessible from the ground state. Here we present direct observations of high-order coherences in gallium arsenide quantum wells, achieved using two-dimensional multiple-quantum spectroscopy methods in which up to seven successive light fields were used. The measurements were made possible by the combination of a reconfigurable spatial beam-shaper that formed multiple beams in specified geometries and a spatiotemporal pulse-shaper that controlled the relative optical phases and temporal delays among pulses in all the beams. The results reveal triexciton coherences (correlations of three excitons or six particles), whose existence was not obvious because the third exciton spin is unpaired, and the values of their coherence times and binding energies. Rephasing of biexcitons, triexcitons and unbound two-exciton coherences was demonstrated. We also determined that there are no significant unbound correlations of three excitons and no bound or unbound four-exciton (eight-particle) correlations. Thus, the limits, as well as the properties, of many-body correlations in this system were revealed. The measurement methods open a new window into high-order many-body interactions in materials and molecules, and the present results should guide ongoing work on first-principles calculations of electronic interactions in semiconductor nanostructures.