Extreme high-frequency BL Lacs (EHBL) feature their synchrotron peak of the broad-band spectral energy distribution (SED) at νs ≥ 1017 Hz. The BL Lac object 1ES 2344+514 was included in the EHBL family because of its impressive shift of the synchrotron peak in 1996. During the following years, the source appeared to be in a low state without showing any extreme behaviours. In 2016 August, 1ES 2344+514 was detected with the ground-based γ-ray telescope FACT during a high γ-ray state, triggering multiwavelength (MWL) observations. We studied the MWL light curves of 1ES 2344+514 during the 2016 flaring state, using data from radio to very-high-energy (VHE) γ-rays taken with OVRO, KAIT, KVA, NOT, some telescopes of the GASP-WEBT collaboration at the Teide, Crimean, and St. Petersburg observatories, Swift-UVOT, Swift-XRT, Fermi-LAT, FACT, and MAGIC. With simultaneous observations of the flare, we built the broad-band SED and studied it in the framework of a leptonic and a hadronic model. The VHE γ-ray observations show a flux level of 55 per cent of the Crab Nebula flux above 300 GeV, similar to the historical maximum of 1995. The combination of MAGIC and Fermi-LAT spectra provides an unprecedented characterization of the inverse-Compton peak for this object during a flaring episode. The Γ index of the intrinsic spectrum in the VHE γ-ray band is 2.04 ± 0.12stat ± 0.15sys. We find the source in an extreme state with a shift of the position of the synchrotron peak to frequencies above or equal to 1018 Hz.