  Extreme High-frequency BL~Lacs (EHBL) feature their synchrotron peak of the
broadband spectral energy distribution (SED) at $\nu_{\rm s} \geq
$10$^{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,
\textit{Swift}-UVOT, \textit{Swift}-XRT, \textit{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 300\,GeV, similar to the historical maximum of 1995. The combination of
MAGIC and \textit{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
$2.04\pm0.12_{\rm stat}\pm0.15_{\rm sys}$. 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
