It should be possible in theory to match the energy density of gasoline. You'd need to come up with a battery chemistry which takes in oxygen when it discharges and produces oxygen when it charges. Effectively, reversible combustion (or fuel cells). It's a tough problem but I don't think there's anything that fundamentally prevents it in theory.

Going much beyond it probably requires a nuclear process. You can store energy in certain isotopes by bombarding them with x-rays or gamma rays of a precise frequency. Stored energy can then be released with a similar process. In theory, it's possible to build a battery that uses this approach. In practice... to call it a difficult engineering problem would be a severe understatement.