We observed nine primary transits of the hot Jpiter TrES-3b in several optical and near-UV photometric bands from 2009 June to 2012 April in an attempt to detect its magnetic field. Vidotto, Jardine and Helling suggest that the magnetic field of TrES-3b can be constrained if its near-UV light curve shows an early ingress compared to its optical light curve, while its egress remains unaffected. Predicted magnetic field strengths of Jupiter-like planets should range between 8G and 30G. Using these magnetic field values and an assumed B* of 100G, the Vidotto et al. method predicts a timing difference of 5-11min. We did not detect an early ingress in our three nights of near-UV observations, despite an average cadence of 68 s and an average photometric precision of 3.7mmag. However, we determined an upper limit of TrES-3b's magnetic field strength to range between 0.013 and 1.3G (for a 1-100G magnetic field strength range for the host star, TrES-3) using a timing difference of 138s derived from the Nyquist-Shannon sampling theorem. To verify our results of an abnormally small magnetic field strength for TrES-3b and to further constrain the techniques of Vidotto et al., we propose future observations of TrES-3b with other platforms capable of achieving a shorter near-UV cadence. We also present a refinement of the physical parameters of TrES-3b, an updated ephemeris and its first published near-UV light curve. We find that the near-UV planetary radius of Rp=1.386+0.248-0.144RJup is consistent with the planet's optical radius.