There must be some limit of how much energy can be packed into a certain space, before it either becomes ridiculously unstable or dangerous. I do wonder where that limit is?

E=mc^2.

The question is not how much energy there is in a thing, but how to get it out, usefully!

When it comes to using lithium ions as energy vectors of storage, a lithium ion has an electro-potential of around -3V and the ion it latches on to will be typically +1V or so, so 4V in total. This represents an energy of 4eV per atom, thus 4J per Coulomb.

There are ~96,000 Coulombs in one mole of charge (this is called the 'Faraday Constant', in fact) and thus ~400kJ per mole of lithium, which is around 7g/mole.

1kWh is 3.6MJ so 1kWh requires 9 moles of lithium, around 56g of it.

A 60kWh battery is therefore 'actually' using 3.3kg of lithium, flowing back and forth between its electrodes in the electrolyte. Of course, those latter parts, and all the stuff that holds it and connects it, is the major mass. The question of how much can you pack into a battery is therefore how small can you make 'those'?

(I've rushed that out from memory, if I have made an error in the recollection or calculation please let me know.)