Upper limit for superconducting transition temperature in electron-phonon superconductors: very strong coupling

We present a brief review of some recent work on the problem of highest achievable temperature of superconducting transition $T_c$ in electron-phonon systems. The discovery of record-breaking values of $T_c$ in quite a number of hydrides under high pressure was an impressive demonstration of capabilities of electron-phonon mechanism of Cooper pairing. This lead to an increased interest on possible limitations of Eliashberg-McMillan theory as the main theory of superconductivity in a system of electrons and phonons. We shall consider some basic conclusions following from this theory and present some remarks on the limit of very strong electron-phonon coupling. We shall discuss possible limitations on the value of the coupling constant related to possible lattice and specific heat instability and conclude that within the stable metallic phase the effective pairing constant may acquire very large values. We discuss some bounds for $T_c$ derived in the strong coupling limit and propose an elementary estimate of an upper limit for $T_c$, expressed via combination of fundamental physical constants. Finally we also briefly discuss some pessimistic estimates for $T_c$ of metallic hydrogen obtained in ``jellium'' model.