We study the band gap of finite armchair graphene nanoribbons (7-AGNRs) on Au(111) through scanning tunneling microscopy/spectroscopy combined with density functional theory calculations. The band gap of 7-AGNRs with lengths of 8 nm and more is converged to within 50 meV of its bulk value of , while the band gap opens by several hundred meV in very short 7-AGNRs. We demonstrate that even an atomic defect, such as the addition of one hydrogen atom at the termini, has a significant effect - in this case, lowering the band gap. The effect can be captured in terms of a simple analytical model by introducing an effective "electronic length".