The discovery of transiting planets around bright stars holds the potential to greatly enhance our understanding of planetary atmospheres. In this work we present the search for transits of HD 168443b, a massive planet orbiting the bright star HD 168443 (V = 6.92) with a period of 58.11 days. The high eccentricity of the planetary orbit (e = 0.53) significantly enhances the a priori transit probability beyond that expected for a circular orbit, making HD 168443 a candidate for our ongoing Transit Ephemeris Refinement and Monitoring Survey. Using additional radial velocities from Keck High Resolution Echelle Spectrometer, we refined the orbital parameters of this multiplanet system and derived a new transit ephemeris for HD 168443b. The reduced uncertainties in the transit window make a photometric transit search practicable. Photometric observations acquired during predicted transit windows were obtained on three nights. Cerro Tololo Inter-American Observatory 1.0 m photometry acquired on 2010 September 7 had the required precision to detect a transit but fell just outside of our final transit window. Nightly photometry from the T8 0.8 m automated photometric telescope at Fairborn Observatory, acquired over a span of 109 nights, demonstrates that HD 168443 is constant on a timescale of weeks. Higher-cadence photometry on 2011 April 28 and June 25 shows no evidence of a transit. We are able to rule out a non-grazing transit of HD 168443b.