Context. Giant radio galaxies (GRGs, or colloquially giants) are the Universes largest structures generated by individual galaxies. They comprise synchrotron-radiating active galactic nucleus ejecta and attain cosmological (megaparsec-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 DR2 by subtracting compact sources and performing multi-scale CLEAN de-convolutions at 60 and 90 resolution. The resulting images constitute the most sensitive survey yet for radio galaxy lobes, whose diffuse 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 have discovered Alcyoneus, a low-excitation radio galaxy with a projected proper length lp=4.99±0.04 Mpc. Both its jets and lobes are detected at very high significance, and the SDSS-based identification of the host, at spectroscopic redshift zspec=0.24674±6-105, is unambiguous. The total luminosity density at v=144 MHz is Lv =8±1-1025 W Hz1, 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±0.4-1011M and a super-massive black hole mass M=4±2-108 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 filament of the Cosmic Web. Through a new Bayesian model for radio galaxy lobes in three dimensions, we estimate the pressures in the megaparsec-cubed-scale northern and southern lobes to be Pmin,1 =4.8±0.3-10 16 Pa and Pmin,2=4.9±0.6-1016 Pa, respectively. The corresponding magnetic field strengths are Bmin,1=46±1 pT and Bmin,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 lp=4.99±0.04 Mpc. The true proper length is at least lmin=5.04±0.05 Mpc. Beyond geometry, Alcyoneus and its host are suspiciously ordinary: the total low-frequency luminosity density, stellar mass, and super-massive 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 low-density environment remains a possible explanation. The source resides in a filament of the Cosmic Web, with which it might have significant thermodynamic interaction. The pressures in the lobes are the lowest hitherto found, and Alcyoneus therefore represents the most promising radio galaxy yet to probe the warm" hot inter-galactic medium.
