Magnetic tunnel junctions with perpendicular anisotropy form the basis of the spin-transfer torque magnetic random-access memory (STT-MRAM), which is non-volatile, fast, dense, and has quasi-infinite write endurance and low power consumption. Based on density functional theory (DFT) calculations, we propose an alternative design of magnetic tunnel junctions comprising Fe(n)Co(m)Fe(n)/MgO storage layers with greatly enhanced perpendicular magnetic anisotropy (PMA) up to several mJ/m2, leveraging the interfacial perpendicular anisotropy of Fe/MgO along with a stress-induced bulk PMA discovered within bcc Co. This giant enhancement dominates the demagnetizing energy when increasing the film thickness. The tunneling magnetoresistance (TMR) estimated from the Julliere model is comparable with that of the pure Fe/MgO case. We discuss the advantages and pitfalls of a real-life fabrication of the structure and propose the Fe(3ML)Co(4ML)Fe(3ML) as a storage layer for MgO-based STT-MRAM cells. The large PMA in strained bcc Co is explained in the framework of Brunos model by the MgO-imposed strain and consequent changes in the energies of dyz and dz2 minority-spin bands.