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We describe directed searches for continuous gravitational waves (GWs) in data from the sixth Laser Interferometer Gravitational-wave Observatory (LIGO) science data run. The targets were nine young supernova remnants not associated with pulsars; eight of the remnants are associated with non-pulsing suspected neutron stars. One target's parameters are uncertain enough to warrant two searches, for a total of 10. Each search covered a broad band of frequencies and first and second frequency derivatives for a fixed sky direction. The searches coherently integrated data from the two LIGO interferometers over time spans from 5.3–25.3 days using the matched-filtering -statistic. We found no evidence of GW signals. We set 95% confidence upper limits as strong (low) as 4 × 10⁻²⁵ on intrinsic strain, 2 × 10⁻⁷ on fiducial ellipticity, and 4 × 10⁻⁵ on r-mode amplitude. These beat the indirect limits from energy conservation and are within the range of theoretical predictions for neutron-star ellipticities and r-mode amplitudes.