  The BL Lac 1ES 2344+514 is known for temporary extreme properties (e.g., a
shift of the synchrotron SED peak energy $\nu_{synch,p}$ above 1keV). While
those extreme states were so far observed only during high flux levels,
additional multi-year observing campaigns are required to achieve a coherent
picture. Here, we report the longest investigation of the source from radio to
VHE performed so far, focusing on a systematic characterisation of the
intermittent extreme states. While our results confirm that 1ES 2344+514
typically exhibits $\nu_{synch,p}>$1keV during elevated flux periods, we also
find periods where the extreme state coincides with low flux activity. A strong
spectral variability thus happens in the quiescent state, and is likely caused
by an increase of the electron acceleration efficiency without a change in the
electron injection luminosity. We also report a strong X-ray flare (among the
brightest for 1ES 2344+514) without a significant shift of $\nu_{synch,p}$.
During this particular flare, the X-ray spectrum is among the softest of the
campaign. It unveils complexity in the spectral evolution, where the common
harder-when-brighter trend observed in BL Lacs is violated. During a low and
hard X-ray state, we find an excess of the UV flux with respect to an
extrapolation of the X-ray spectrum to lower energies. This UV excess implies
that at least two regions contribute significantly to the
infrared/optical/ultraviolet/X-ray emission. Using the simultaneous MAGIC,
XMM-Newton, NuSTAR, and AstroSat observations, we argue that a region possibly
associated with the 10 GHz radio core may explain such an excess. Finally, we
investigate a VHE flare, showing an absence of simultaneous variability in the
0.3-2keV band. Using a time-dependent leptonic modelling, we show that this
behaviour, in contradiction to single-zone scenarios, can instead be explained
by a two-component model.
