Models and observations suggest that both power and effects of AGN feedback should be maximised in hyper-luminous (L_Bol>10^47 erg/s) quasars, i.e. objects at the brightest end of the AGN luminosity function. We present the first results of a multi-wavelength observing program, focusing on a sample of WISE/SDSS selected hyper-luminous (WISSH) broad-line quasars at z~1.5-5. The WISSH quasars project has been designed to reveal the most energetic AGN-driven outflows, estimate their occurrence at the peak of quasar activity and extend the study of correlations between outflows and nuclear properties up to poorly-investigated extreme AGN luminosities (L_Bol~10^47 -10^48 erg/s). We present NIR, long-slit LBT/LUCI1 spectroscopy of five WISSH quasars at z~2.3-3.5 showing prominent [OIII] emission lines with broad (FWHM~1200-2200 km/s) and skewed profiles. The luminosities of the broad [OIII] wings are the highest measured so far (L_[OIII]^broad >~5x10^44 erg/s) and reveal the presence of powerful ionised outflows with mass outflow rates Mdot >~1700 M_Sun/yr and kinetic powers Edot >~10^45 erg/s. Although these estimates are affected by large uncertainties, due to the use of [OIII] as tracer of ionized outflows and the very basic outflow model we assume, these results suggest that the AGN is highly efficient in pushing outwards large amounts of ionised gas in our targets. The mechanical outflow luminosities for WISSH quasars correspond to higher fractions (~1-3%) of L_Bol than those derived for lower L_Bol AGN. Our targets host very massive (M_BH>~2x10^9 M_Sun) black holes which are still accreting at a high rate (i.e. a factor of ~0.4-3 of the Eddington limit). These findings demonstrate that WISSH quasars offer the opportunity of probing the extreme end of both luminosity and SMBH mass functions and revealing powerful ionised outflows able to affect the evolution of their host galaxies. ; Comment: 18 pages, 11 figures, A&A accepted