We report first results from a large project to measure black hole (BH) mass in high accretion rate active galactic nuclei (AGNs). Such objects may be different from other AGNs in being powered by slim accretion disks and showing saturated accretion luminosities, but both are not yet fully understood. The results are part of a large reverberation mapping (RM) campaign using the 2.4m Shangri-La telescope at the Yunnan Observatory in China. The goals are to investigate the gas distribution near the BH and the properties of the central accretion disks, to measure BH mass and Eddington ratios, and to test the feasibility of using such objects as a new type of cosmological candles. The paper presents results for three objects, Mrk 335, Mrk 142, and IRAS F12397+3333, with H{beta} time lags relative to the 5100{AA} continuum of 10.6_-2.9_^+1.7^, 6.4_-2.2_^+0.8^ and 11.4_-1.9_^+2.9^ days, respectively. The corresponding BH masses are (8.3_-3.2_^+2.6^)x10^6^M_{sun}_, (3.4_-1.2_^+0.5^)x10^6^M_{sun}_, and (7.5_-4.1_^+4.3^x10^6^M_{sun}_, and the lower limits on the Eddington ratios are 0.6, 2.3, and 4.6 for the minimal radiative efficiency of 0.038. Mrk 142 and IRAS F12397+333 (extinction corrected) clearly deviate from the currently known relation between H{beta} lag and continuum luminosity. The three Eddington ratios are beyond the values expected in thin accretion disks and two of them are the largest measured so far among objects with RM-based BH masses. We briefly discuss implications for slim disks, BH growth, and cosmology.