We report on an all-sky search for periodic gravitational waves in the frequency band 20-475 Hz and with a frequency time derivative in the range of [-1.0, +0.1] x 10(-8) Hz/s. Such a signal could be produced by a nearby spinning and slightly nonaxisymmetric isolated neutron star in our galaxy. This search uses the data from Advanced LIGO''s first observational run, O1. No periodic gravitational wave signals were observed, and upper limits were placed on their strengths. The lowest upper limits on worst-case (linearly polarized) strain amplitude h(0) are similar to 4 x 10(-25) near 170 Hz. For a circularly polarized source (most favorable orientation), the smallest upper limits obtained are similar to 1.5 x 10(-25). These upper limits refer to all sky locations and the entire range of frequency derivative values. For a population-averaged ensemble of sky locations and stellar orientations, the lowest upper limits obtained for the strain amplitude are similar to 2.5 x 10(-25).