Are there any battery chemistries that can be fabricated at home?
There are - or were - but this may not be the answer you want. 

If you look at the September 1901 issue of "Model Engineer and Amateur Electrician" (now just "Model Engineer") you'll find full construction drawings for a 2-cell battery with (according to the article) a low internal resistance, using zinc and carbon electrodes, and "four ounces of chromic acid or bichromate of potash dissolved in 12ounces of water". All common materials, readily available from your local chemist (pharmacy) - at least, when it was written.

The battery had one interesting feature : to switch it on or off, you lowered the electrodes into the solution, or raised them.

To give some perspective, these instructions come from the heroic days of the magazine, when it was perfectly normal to build your own motorbike, casting and machining the engine's cylinder yourself. 

This can be seen from the assembly instructions : "First well amalgamate the zincs by dipping them in dilute sulphuric acid, and then rubbing them over with mercury (use a piece of rag at the end of a stick for this)" - see the emphasis on safety here? 

Lifetime? "If the lamp does not take more than 1/3 ampere, it should remain at full brilliancy for two hours right off", and "Renew solution when current weakens". (It's a primary cell, not rechargeable).

You would need to adapt this basic design to your form factor... though a slim tray full of LiPo cells is probably a lot easier.