1ES 1959+650 is a bright TeV high-frequency-peaked BL Lac object exhibiting interesting features like "orphan" TeV flares and a broad emission in the high-energy regime, that are difficult to interpret using conventional one-zone Synchrotron Self-Compton (SSC) scenarios. We report the results from the Major Atmospheric Gamma Imaging Cherenkov (MAGIC) observations in 2016 along with the multi-wavelength data from the Fermi Large Area Telescope (LAT) and Swift instruments. MAGIC observed 1ES 1959+650 with different emission levels in the very-high-energy (VHE, E >100 GeV) gamma-ray band during 2016. In the long-term data, the X-ray spectrum becomes harder with increasing flux and a hint of a similar trend is also visible in the VHE band. An exceptionally high VHE flux reaching ~ 3 times the Crab Nebula flux was measured by MAGIC on the 13th, 14th of June and 1st July 2016 (the highest flux observed since 2002). During these flares, the high-energy peak of the spectral energy distribution (SED) lies in the VHE domain and extends up to several TeV. The spectrum in the gamma-ray (both Fermi-LAT and VHE bands) and the X-ray bands are quite hard. On 13th June and 1st July 2016, the source showed rapid variations of the VHE flux within timescales of less than an hour. A simple one-zone SSC model can describe the data during the flares requiring moderate to high values of the Doppler factors (>=30-60). Alternatively, the high-energy peak of the SED can be explained by a purely hadronic model attributed to proton-synchrotron radiation with jet power L_{jet}~10^{46} erg/s and under high values of the magnetic field strength (~100 G) and maximum proton energy (~few EeV). Mixed lepto-hadronic models require super-Eddington values of the jet power. We conclude that it is difficult to get detectable neutrino emission from the source during the extreme VHE flaring period of 2016.