We describe directed searches for continuous gravitational waves in data from the sixth 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 targets parameters are uncertain enough to warrant two searches, for a total of ten. 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 F-statistic. We found no credible gravitational-wave signals. We set 95% confidence upper limits as strong (low) as $4times10^{-25}$ on intrinsic strain, $2times10^{-7}$ on fiducial ellipticity, and $3times10^{-6}$ 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.
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