We report results of a deep all-sky search for periodic gravitational waves from isolated neutron stars in data from the S6 LIGO science run. The search was possible thanks to the computing power provided by the volunteers of the Einstein@Home distributed computing project. We find no significant signal candidate and set the most stringent upper limits to date on the amplitude of gravitational wave signals from the target population. At the frequency of best strain sensitivity, between 170.5 and 171 Hz we set a 90% confidence upper limit of 5.5×10−25, while at the high end of our frequency range, around 505 Hz, we achieve upper limits ≃10−24. At 230 Hz we can exclude sources with ellipticities greater than 10−6 within 100 pc of Earth with fiducial value of the principal moment of inertia of 1038  kg m2. If we assume a higher (lower) gravitational wave spin-down we constrain farther (closer) objects to higher (lower) ellipticities.