What's the highest theoretical energy density for a chemical battery?
I don't know the actual answer to this question, but I know a least upper bound to the answer, and a means of figuring out the real answer.

Battery scientists have a metric called maximum theoretical specific energy; you can read about the definition in Advanced Batteries by Robert Huggins. Right now, the most energy dense batteries you can buy are lithium ion, which are in the 100-200 Wh/kg range. I don't know what the best battery is, but later in the book, Huggins shows calculations that indicate that Li/CuCl2 cells have an MTSE of 1166.4 Wh/kg. (5x the capacity of current batteries!)

We know that the highest MTSE is at least 1166.4 Wh/kg; you could use his method to calculate the same value for other chemistries, but the search space is pretty large.

I've also seen references on the internet to Li/O2 and Al/O2 batteries with MTSE of 2815 and 5200 Wh/kg, respectively. Not sure how credible those references are. Later references, like this 2008 article in the Journal of the Electrochemical Society, suggest that the MTSE for a Li/O2 cell is around 1400 Wh/kg.