[en] We present rough variational upper-bound estimates of some low-lying bound-state energy levels of two-electron systems, including H^-, He, and Li⁺, in magnetic fields B above 10¹⁰ G, a regime that is relevant to recent astrophysical studies of pulsars. The simple trial wave functions that we use in the Rayleigh-Ritz calculations consist of single antisymmetrized products of single-particle orbitals which are predominantly magnetic in their spatial character. Most importantly, our results indicate that in the regime 10¹⁰ approximately-less-than B approximately-less-than 10¹³ G the H^- ion has at least one singlet and one triplet bound state; at 2 times 10¹² G, e. g., the ionization energy of the triplet ground state of H^- is greater than 12 eV. (The number of bound states of H^- in this regime of B may be much greater; moreover, the number of levels and their depths is expected to increase monotonically with B.) These results appear to contradict a recent indication that there might not be any bound states of H^- above 10¹⁰ G. For B → infinity, we also present the approximate leading asymptotic terms for the total (nonrelativistic) energies and corresponding ionization energies of some low-lying states of two-electron systems with nuclear charge Z greater than or equal to 2