What batteries are available for powering outdoor LED lights?
The next size up from AA is C.  The next size up from C is D.  You can get both of those in rechargeable, and a sufficient stack of them will cover you just fine.
Over in the lithium world, we have the common 18650 size which about the size of two C-batteries with more energy storage for their size.
Do not underestimate the design challenges of working with LEDs.  Your best bet is to use lights designed to a standard voltage such as 12V &quot;LED strips&quot; and then use 12V batteries.  For instance three lithiums or 10 NiMH batteries will give 12V. Most 12V LED assemblies are tolerant of 11-14V, since those are voltages that will come off a car battery at different times.