Stellar variability in Gaia DR3 I. Three-band photometric dispersions for 145 million sources

Context. The unparalleled characteristics of Gaia photometry in terms of calibration, stability, time span, dynamic range, full-sky coverage, and complementary information make it an excellent choice to study stellar variability. Aims. We aim to measure the photometric dispersion in the G, GBP, and GRP bands of the 145 677 450 third Gaia data release (DR3) five-parameter sources with G <= 17 mag and GBP - GRP between -1.0 and 8.0 mag. We will use that unbiased sample to analyze stellar variability in the Milky Way (MW), Large Magellanic Cloud (LMC), and Small Magellanic Cloud (SMC). Methods. For each band we convert from magnitude uncertainties to observed photometric dispersions, calculate the instrumental component as a function of apparent magnitude and color, and use it to transform the observed dispersions into the astrophysical ones: s(G), s(GBP), and s(GRP). We give variability indices in the three bands for the whole sample indicating whether the objects are non-variable, marginally variable, or clearly so. We use the subsample established by Rimoldini and collaborators with light curves and variability types to calibrate our results and establish their limitations. Results. The position of an object in the dispersion-dispersion planes can be used to constrain its variability type, a direct application of these results. We use information from the MW, LMC, and SMC color-absolute magnitude diagrams (CAMDs) to discuss variability across the Hertzsprung-Russell diagram. White dwarfs and B-type subdwarfs are more variable than main sequence (MS) or red clump (RC) stars, with a flat distribution in sG up to 10 mmag and with variability decreasing for the former with age. The MS region in the Gaia CAMD includes a mixture of populations from the MS itself and from other evolutionary phases. Its sG distribution peaks at low values (similar to 1-2 mmag) but it has a large tail dominated by eclipsing binaries, RR Lyrae stars, and young stellar objects. RC stars are characterized by little variability, with their s(G) distribution peaking at 1 mmag or less. The stars in the pre-main-sequence (PMS) region are highly variable, with a power law distribution in sG with slope 2.75 and a cutoff for values lower than 7 mmag. The luminous red stars region of the Gaia CAMD has the highest variability, with its extreme dominated by AGB stars and with a power law in s(G) with slope similar to 2.2 that extends from there to a cutoff of 7 mmag. We show that our method can be used to search for LMC Cepheids. We analyze four stellar clusters with O stars (Villafranca O-016, O-021, O-024, and O-026) and detect a strong difference in s(G) between stars that are already in the MS and those that are still in the PMS.