The magnetoelectric properties of exchange-coupled Ni/CoFeB-based composite multiferroic microstructures are investigated. The strength and sign of the magnetoelastic effect are found to be strongly correlated with the ratio between the thicknesses of two magnetostrictive materials. In cases where the thickness ratio deviates significantly from one, the magnetoelastic behavior of the multiferroic microstructures is dominated by the thicker layer, which contributes more strongly to the observed magnetoelastic effect. More symmetric structures with a thickness ratio equal to one show an emergent interfacial behavior which cannot be accounted for simply by summing up the magnetoelastic effects occurring in the two constituent layers. This aspect is clearly visible in the case of ultrathin bilayers, where the exchange coupling drastically affects the magnetic behavior of the Ni layer, making the Ni/CoFeB bilayer a promising next-generation synthetic magnetic system entirely. This study demonstrates the richness and high tunability of composite multiferroic systems based on coupled magnetic bilayers compared to their single magnetic layer counterparts. Furthermore, because of the compatibility of CoFeB with present magnetic tunnel junction-based spintronic technologies, the reported findings are expected to be of great interest for the development of ultralow-power magnetoelectric memory devices.