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This article presents the results of experimental studies of the impact of electrode material and the effect of nanoscale film thickness on the resistive switching in forming-free nanocrystalline ZnO films grown by pulsed laser deposition. It was demonstrated that the nanocrystalline ZnO film with TiN, Pt, ZnO:In, and ZnO:Pd bottom electrodes exhibits a nonlinear bipolar effect of forming-free resistive switching. The sample with Pt showed the highest resistance values RHRS and RLRS and the highest value of Uset = 2.7 ± 0.4 V. The samples with the ZnO:In and ZnO:Pd bottom electrode showed the lowest Uset and Ures values. An increase in the number of laser pulses from 1000 to 5000 was shown to lead to an increase in the thickness of the nanocrystalline ZnO film from 7.2 ± 2.5 nm to 53.6 ± 18.3 nm. The dependence of electrophysical parameters (electron concentration, electron mobility, and resistivity) on the thickness of the forming-free nanocrystalline ZnO film for the TiN/ZnO/W structure was investigated. The endurance test and homogeneity test for TiN/ZnO/W structures were performed. The structure Al2O3/TiN/ZnO/W with a nanocrystalline ZnO thickness 41.2 ± 9.7 nm was shown to be preferable for the manufacture of ReRAM and memristive neuromorphic systems due to the highest value of RHRS/RLRS = 2307.8 ± 166.4 and low values of Uset = 1.9 ± 0.2 V and Ures = −1.3 ± 0.5 V. It was demonstrated that the use of the TiN top electrode in the Al2O3/TiN/ZnO memristor structure allowed for the reduction in Uset and Ures and the increase in the RHRS/RLRS ratio. The results obtained can be used in the manufacturing of resistive-switching nanoscale devices for neuromorphic computing based on the forming-free nanocrystalline ZnO oxide films.