Beam-helicity asymmetries for the gamma+p -> pi+ + pi- + p reaction have been measured for center-of-mass energies between 1.35 GeV and 2.30 GeV at Jefferson Lab with the CEBAF Large Acceptance Spectrometer using circularly polarized tagged photons. The beam-helicity asymmetries vary with kinematics and exhibit strong sensitivity to the dynamics of the reaction, as demonstrated in the comparison of the data with results of various phenomenological model calculations. These models currently do not provide an adequate description of the data over the entire kinematic range covered in this experiment. Additional polarization observables are accessible in an upcoming experiment at Jefferson Lab with polarized beam and target.