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We present the results of an all-sky search for continuous gravitational waves in the public LIGO O3 data. The search covers signal frequencies $20$ Hz $\leq f \leq 800$ Hz and a spin-down range down to $-2.6\times 10^{-9}$ Hz s$^{-1}$1, motivated by detectability studies on synthetic populations of Galactic neutron stars. This search is the most sensitive all-sky search to date in this frequency/spin-down region. The initial search was performed using the first half of the public LIGO O3 data (O3a), utilizing Graphical Processing Units provided in equal parts by the volunteers of the Einstein@Home computing project and by the ATLAS cluster. After a hierarchical follow-up in seven stages, 12 candidates remain. Six are discarded at the eighth stage, by using the remaining O3 LIGO data (O3b). The surviving six can be ascribed to continuous-wave fake signals present in the LIGO data for validation purposes. We recover these fake signals with very high accuracy with our last stage search, which coherently combines all O3 data. Based on our results, we set upper limits on the gravitational wave amplitude $h_0$, and translate these in upper limits on the neutron star ellipticity and on the $r$-mode amplitude. The most stringent upper limits are at $203$ Hz, with $h_0=8.1 \times 10^{-26}$ at the 90% confidence level. Our results exclude neutron stars rotating faster than $5$ ms with ellipticities greater than $5\times 10^{-8} \left[{d\over{100~\textrm{pc}}}\right]$ within a distance $d$ from Earth and $r$-mode amplitudes $α\geq 10^{-5} \left[{d\over{100~\textrm{pc}}}\right]$ for neutron stars spinning faster than $150$ Hz.