Context. Giant radio galaxies (GRGs, or colloquially 'giants') are the Universe's largest structures generated by individual galaxies. They comprise synchrotron-radiating AGN ejecta and attain cosmological (Mpc-scale) lengths. However, the main mechanisms that drive their exceptional growth remain poorly understood. Aims. To deduce the main mechanisms that drive a phenomenon, it is usually instructive to study extreme examples. If there exist host galaxy characteristics that are an important cause for GRG growth, then the hosts of the largest GRGs are likely to possess them. Similarly, if there exist particular large-scale environments that are highly conducive to GRG growth, then the largest GRGs are likely to reside in them. For these reasons, we aim to perform a case study of the largest GRG available. Methods. We reprocessed the LOFAR Two-metre Sky Survey (LoTSS) DR2 by subtracting compact sources and performing multiscale CLEAN deconvolution at 60 and 90 resolution. The resulting images constitute the most sensitive survey yet for radio galaxy lobes, whose di use nature and steep synchrotron spectra have allowed them to evade previous detection attempts at higher resolution and shorter wavelengths. We visually searched these images for GRGs. Results. We discover Alcyoneus, a low-excitation radio galaxy with a projected proper length l p = 4.99 ± 0.04 Mpc. Its jets and lobes are all four detected at very high signi cance, and the SDSS-based identi cation of the host, at spectroscopic redshift z spec = 0.24674 ± 6•10 −5 , is unambiguous. The total luminosity density at ν = 144 MHz is L ν = 8±1•10 25 W Hz −1 , which is below-average, though near-median (percentile 45 ± 3%), for GRGs. The host is an elliptical galaxy with a stellar mass M = 2.4 ± 0.4 • 10 11 M and a supermassive black hole mass M • = 4 ± 2 • 10 8 M , both of which tend towards the lower end of their respective GRG distributions (percentiles 25±9% and 23±11%). The host resides in a lament of the Cosmic Web. Through a new Bayesian model for radio galaxy lobes in three dimensions, we estimate the pressures in the Mpc 3 -scale northern and southern lobe to be P min,1 = 4.8 ± 0.3 • 10 −16 Pa and P min,2 = 4.9±0.6•10 −16 Pa, respectively. The corresponding magnetic eld strengths are B min,1 = 46±1 pT and B min,2 = 46±3 pT. Conclusions. We have discovered what is in projection the largest known structure made by a single galaxy -a GRG with a projected proper length l p = 4.99 ± 0.04 Mpc. The true proper length is at least l min = 5.04 ± 0.05 Mpc. Beyond geometry, Alcyoneus and its host are suspiciously ordinary: the total low-frequency luminosity density, stellar mass and supermassive black hole mass are all lower than, though similar to, those of the medial GRG. Thus, very massive galaxies or central black holes are not necessary to grow large giants, and, if the observed state is representative of the source over its lifetime, neither is high radio power. A lowdensity environment remains a possible explanation. The source resides in a lament of the C...