Organic radicals are promising building blocks for molecular spintronics. Little is known about the role of unpaired electrons for electron transport at the single molecule level. Here, we examine the impact of magnetic fields on electron transport in single oligo(p-phenyleneethynylene) (OPE)-based radical molecular junctions which are formed with a mechanically controllable break junction technique at a low temperature of 4.2 K. Surprisingly huge positive magnetoresistances (MRs) of 16 to 287 % are visible for a magnetic field of 4 T, and the values are at least one order of magnitude larger than those of the analogous pristine OPE (2-4 %). Rigorous analysis of the MR, of current-voltage and inelastic electron tunneling spectroscopy measurements reveal an effective reduction of the electronic coupling between the current-carrying molecular orbital and the electrodes with increasing magnetic field. We suggest that the large MR for the single radical molecular junctions might be ascribed to a loss of phase coherence of the charge carriers induced by the magnetic field. Although further investigations are required to reveal the mechanism underlying the strong MR, our findings provide a potential approach for tuning charge transport in metal-molecule junctions with organic radicals.