What is the longest lasting rechargable battery?

It depends on the needs of the device vs the strengths and weaknesses of the battery. NiMh is cheap and can supply lots of current, have a flat discharge curve, and can be fast charged but determining their state of charge (i.e. remaining power) is hard . Lithium Ion don’t fast charge very well, but their full point is easy to find. Lead acid are cheap. Some are good at low temp, some have high self discharge. Some can take overcharge, some can’t. Some can be made to fit odd shapes, some are very stable and can take physical damage. Others burn easily or corrode badly if they leak.

If you are asking what brand then in the NiMH type the Panasonic Eneloop is regarded highly and has low self discharge but are rather expensive. jedoch, other battery when tested perform like the Eneloop or better in some respects or are cheaper. Like the above, the answer varies according to your needs. If you don’t need low self discharge or high current output, it might be unwise to use a more expensive battery. And the reverse.