I'm a big fan of modularity and it's requirement, standardization. Lately, I've been learning a little bit about "lithium ion" (there are many types) 18650 cells. They are prolific in laptop batteries, power tool batteries, recharchable flashlights, USB power banks, and more. And while it's common place to buy packs that have these, it's rare to buy 18650 cells directly.
They are not nearly as small as AA, but not hefty, either. The power density is quite decent and varies greatly, but generally speaking they are worth consider. 18650 describes the size, 18mm by 65.0mm (maybe the 0 has another meaning, but I think not). They are generally about 3.7 volts nominal.
Anyway, this isn't meant to be an introduction to electronics course or give you lots of tech specs on the batteries. Generally, you can expect to find between 2,000 and 3,500 milliamp hour batteries, depending on the quality and amount of amps it can safely push out. More amps, lower amp-hours. Pretty straight forward. There's also protected cells that have some protection circuitry and unprotected. There's button top and flat top. All details you'll have to work out.
What I've done is found a flashlight that takes 18650 cells directly. I've also found a USB power bank that takes them directly. I could buy sealed units that likely use them and aren't designed to have the batteries replaced, but why? When the batteries finally go bad, I replace them with another. And rather than have internal batteries between many things, I can just swap batteries and charge them as needed. The USB power bank can even charge the batteries for me, so I don't have to take them out of the case as often.
Further, the more things I buy or make that use 18650s, the more comfortable I get with harvesting them from old battery packs. Maybe I have a laptop battery pack with one bad 18650 and a bunch left that are good. These can now become my "consumable" 18650s. It's cheaper, helps reduce waste going to landfills, and gives you more options if you're in a pinch.
Now it's not all rosy. There are major safety concerns with 18650s and lithium batteries in general. While some are very safe, assume they have to be charged and discharged correctly else they can catch fire. They don't normally do this, but it's not some mistake that on Amazon I found a handful of photos with 18650 chargers that had caught fire. I'm guessing most of this comes down to bad design and/or components in the chargers and/or sketchy Chinese cells. It's very easy to buy quality Japanese 18650 cells and personally, I would not mess with anything else unless I already had it on hand. But generally speaking, lithium dangers are a bit understated. On the Grand Tour (not-as-good successor to Top Gear), an electric super car was crashed and the fire burned for 5+ days and could not be put out. Now that's a ton of batteries, but I caution you all the same. Even if you watch Battle Bots now vs years ago, the bots now often catch fire from one issue or another with the batteries and never seemed to 10+ years ago, I'm guessing before they were using lithium as often.
Lithium batteries can be safe when used properly. For starters, don't exceed recommended drain or charging amps, don't charge at too high of a voltage, etc. They're not as forgiving. They're way better on paper than lead acid, Ni-Cd, Ni-MH, etc, but I would call them touchy. I'm no EEVBlog, but I think my caution is legitimate. Further, 18650s are not intended for consumer use, probably largely for these reasons. That said, people use them all the time knowingly and unknowingly. Just be mindful.
As far as which cells I've started with, I opted for the Panasonic NCR18650B. It's protected, high amp hour, Japanese made, and not meant for tons of current. This is great for USB power banks and light-duty flashlights. The higher current cells are probably a bit safer as a rule of thumb in case of shorts, etc. You should do your own research and decide what makes sense for you.
Thanks for reading.