How long does it take to battery decompose?

Answer 1

The question is unanswerable.

"A battery" is pretty vague. Most batteries have multiple parts, many of which are not organic; it's not entirely clear what "decompose" means for something like a piece of metal or a paste of inorganic goop (maybe it's already "decomposed"?).

That said:

Many plastics last effectively forever unless broken up mechanically, because no organisms capable of digesting them exist.

Metal parts are unlikely to be harmed by microbes, but could potentially be subject to chemical corrosion. How fast this is depends on the environment and on precisely what the metal is; some museums have metal artifacts predating recorded history. Most of these are made of relatively inert metals such as gold,but even copper artifacts can last thousands of years.

Besides plastic parts and metal (both of which can be pretty durable), batteries contain some kind of electrolyte. This can be mainly liquid, like in car batteries (where it's sulfuric acid), or some kind of paste like in "dry cells". In either case, this is unlikely to degrade much over time, but will disperse through normal diffusion once the container holding it is breached.