Magneto-optical spectra of a GaAs/AlAs multi-quantum-well sample have been measured in the Faraday configuration at high magnetic fields up to 25 T. These spectra reveal clear excitonic effects on top of the Landau-level structure. The excitonic states are well explained by effective-mass calculations that take into account residual electric fields in the sample and the valence-band mixing in magnetic fields. The results indicate that the Coulomb interaction plays an important role even under very high magnetic fields, in contrast to the common belief that it should be only a weak perturbation to the Landau level. A crossing of the lowest heavy-hole free exciton and the lowest light-hole free exciton is observed at a magnetic field of about 15 T with σ+ polarization, thus achieving a symmetry change in the exciton ground state. The absence of an anticrossing between the light- and heavy-hole exciton ground state indicates the unimportance of exchange effects.