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Interpreting and modelling astronomical catalogues requires an understanding of the catalogues' completeness or selection function: objects of what properties had a chance to end up in the catalogue. Here we set out to empirically quantify the completeness of the overall Gaia DR3 catalogue. This task is not straightforward because Gaia is the all-sky optical survey with the highest angular resolution to date and no consistent ``ground truth'' exists to allow direct comparisons. However, well-characterised deeper imaging enables an empirical assessment of Gaia's $G$-band completeness across parts of the sky. On this basis, we devised a simple analytical completeness model of Gaia as a function of the observed $G$ magnitude and position over the sky, which accounts for both the effects of crowding and the complex Gaia scanning law. Our model only depends on a single quantity: the median magnitude $M_{10}$ in a patch of the sky of catalogued sources with $\texttt{astrometric_matched_transits}$ $\leq 10$. $M_{10}$ reflects elementary completeness decisions in the Gaia pipeline and is computable from the Gaia DR3 catalogue itself and therefore applicable across the whole sky. We calibrate our model using the Dark Energy Camera Plane Survey (DECaPS) and test its predictions against Hubble Space Telescope observations of globular clusters. We find that our model predicts Gaia's completeness values to a few per cent across the sky. We make the model available as a part of the $\texttt{gaiasf}$ Python package built and maintained by the GaiaUnlimited project: $\texttt{https://github.com/gaia-unlimited/gaiaunlimited}$