Extreme High-frequency BL Lacs (EHBL) feature their synchrotron peak of the broadband spectral energy distribution (SED) at {nu}s>=10^17^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 August 2016, 1ES 2344+514 was detected with the ground-based {gamma}-ray telescope FACT during a high {gamma}-ray state, triggering multi-wavelength (MWL) observations. We studied the MWL light curves of 1ES 2344+514 during the 2016 flaring state, using data from radio to VHE {gamma} 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 broadband SED and studied it in the framework of a leptonic and an hadronic model. The VHE {gamma}-ray observations show a flux level of 55% of the Crab Nebula flux above 300GeV, 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 {GAMMA} index of the intrinsic spectrum in the VHE {gamma}-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 10^18^Hz