Aims. The BL Lac 1ES 2344+514 is known for temporary extreme properties characterised by a shift of the synchrotron spectral energy distribution (SED) peak energy νsynch;p above 1 keV. While those extreme states have only been observed during high flux levels thus far, additional multi-year observing campaigns are required to achieve a coherent picture. Here, we report the longest investigation of the source from radio to very high energy (VHE) performed so far, focussing on a systematic characterisation of the intermittent extreme states. Methods.We organised a monitoring campaign covering a 3-year period from 2019 to 2021.Morethan ten instruments participated in the observations in order to cover the emission from radio to VHE. In particular, sensitive X-ray measurements by XMM-Newton, NuSTAR, and AstroSat took place simultaneously with multi-hour MAGIC observations, providing an unprecedented constraint of the two SED components for this blazar. Results. While our results confirm that 1ES 2344+514 typically exhibits νsynch;p > 1 keV 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 in the electron acceleration efficiency without a change in the electron injection luminosity. On the other hand, we also report a strong X-ray flare (among the brightest for 1ES 2344+514) without a significant shift of νsynch;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. By combining Swift-XRT and Swift-UVOT measurements 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 significantly contribute 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 0.3-2 keV band. Using time-dependent leptonic modelling, we show that this behaviour, in contradiction to single-zone scenarios, can instead be explained by a two-component model.
