The most widely used batteries would seem AA, AAA —from 1.2V to 1.5V— and the squared 9V one, also called 6F22. There are numerous options and chargers for them, being available without too much hassle. Anyway, talking about rechargeable batteries in these formats, they are normally based on Ni-MH. This technology should be improved to match the characteristics of Li-Ion, but as of today it's an unfair competition.
Energy density of Ni-MH is about 300 watts-hour/kilogram maximum while Li-ion could surpass 700 w-h/kg. The charge-discharge cycles are more or less the same, although self-discharge rate —one of the most important specifications— is newly superior in Li-ion, with 8% per month at 21ºC. Only LSD (Low Self Discharge) Ni-MH batteries are greater than this, with 75-85% after one year at 20ºC, however the drawback is lesser energy density. All these specifications get worse with more temperature in both technologies.
A Li-ion battery has normally more voltage in the same size. The AA Li-ion counterpart battery has 3.6V, near two volts more. Not always an advantage, but if the goal is to add high voltage Li-ion looks the right choice other time.
Manufacturers do employ Li-ion batteries for portable devices, and they should allow users to change these batteries by themselves with the common system based in springs and cover, with drawings for showing the polarity. Nevertheless, the related industry plays unfair and creates difficulties sealing battery packs, providing no screws or other easy method to renew your battery pack simply changing some batteries as a game for children. There should be legislative regulations because it causes multiple waste and overall environmental impact.
Do you want to know more about these batteries? Try with search terms like "18650 battery" or "Li-ion AA", there's also an intensive market in some areas.