Single walled carbon nanotubes, SWNTs, are either semiconducting, metallic, or quasimetallic. Early theoretical work based on tight-binding models predicted that the band gap of semiconducting carbon nanotubes should increase with decreasing radius and this picture was later confirmed by experiment. However, local-density functional calculations indicate that these models are not accurate for narrow carbon nanotubes, where the effects of curvature can convert nanotubes expected to be semiconductors to metals. This raises the question, what is the largest semiconducting band gap possible in a SWNT? We present results from first-principles calculations for a range of carbon nanotubes with radii between 0.15 and 1 nm. These results indicate that the (4,3) carbon nanotube has the largest band gap of all SWNTs.