We present results of an all-sky search for continuous gravitational waves (CWs), which can be produced by fast-spinning neutron stars with an asymmetry around their rotation axis, using data from the second observing run of the Advanced LIGO detectors. We employ three different semi-coherent methods ($textit{FrequencyHough}$, $textit{SkyHough}$, and $textit{Time-Domain $mathcal{F}$-statistic}$) to search in a gravitational-wave frequency band from 20 to 1922 Hz and a first frequency derivative from $-1times10^{-8}$ to $2times10^{-9}$ Hz/s. None of these searches has found clear evidence for a CW signal, so we present upper limits on the gravitational-wave strain amplitude $h_0$ (the lowest upper limit on $h_0$ is $1.7times10^{-25}$ in the 123-124 Hz region) and discuss the astrophysical implications of this result. This is the most sensitive search ever performed over the broad range of parameters explored in this study.