The BL Lac object 1ES~2344+514 was one of the first sources to be included in the extreme high-peaked BL Lac (EHBL) family. EHBLs are characterised by a broadband spectral energy distribution featuring the synchrotron peak above $\sim 1$ keV. 1ES~2344+514 was detected in very-high-energy (VHE) gamma rays for the first time by the Whipple telescope in 1995. The extreme nature of 1ES~2344+514 in the X-ray band was observed in 1996, when Beppo-SAX detected a large 0.1--10 keV flux variability on timescales of a few hours, during another bright outburst in the X-ray band. This extreme behaviour of the source triggered several multiwavelength campaigns in the following years, during which the source appeared to be in a low state. In August 2016, FACT detected the source in a high state, triggering multiwavelength observations. The combination of MAGIC, FACT, and Fermi-LAT spectra provides an unprecedented characterisation of the inverse-Compton peak for this object during a flaring episode. We collected multiwavelength data, and modelled the broadband emission during this peculiar flaring episode using a leptonic and a hadronic model. The source was in an extreme synchrotron state. The peak frequency obtained from the leptonic model corresponds to a synchrotron peak $\nu_{s}$ at 18~keV. The shift of peak frequency with respect to previous observations is $\sim 2$ orders of magnitude. A harder than usual intrinsic VHE gamma-ray spectrum is observed, with $\Gamma=2.04\pm0.12_{\rm stat}\pm0.15_{\rm sys}$. The leptonic and hadronic models both describe successfully the data, but require a significantly different magnetisation of the emitting zone. Our conclusion is that 1ES~2344+514 belongs to a subcategory of EHBLs, which reveal to be extreme only in some circumstances.