We present the result of searches for gravitational waves from 200 pulsars using data from the first observing run ofthe Advanced LIGO detectors. We find no significant evidence for a gravitational-wave signal from any of thesepulsars, but we are able to set the most constraining upper limits yet on their gravitational-wave amplitudes andellipticities. For eight of these pulsars, our upper limits give bounds that are improvements over the indirect spindownlimit values. For another 32, we are within a factor of 10 of the spin-down limit, and it is likely that some ofthese will be reachable in future runs of the advanced detector. Taken as a whole, these new results improve onprevious limits by more than a factor of two.