Do rechargeable lithium-ion batteries exists in standard sizes like AA, AAA, C or D?

No, they don’t. According to Isidor Buchmann, author of Batteries in a Portable World: A Handbook on Rechargeable Batteries for Non-Engineers, this is due to safety concerns — people might try to charge them in chargers made for other AA batteries, where they might explode. Also, because Li-ion batteries operate 3.7V per cell, rather than the 1.2 to 1.5V of most cell batteries, designing a 1.5V lithium-ion cell would be expensive.

However, you can find good rechargeable AA Nickel-Metal Hydride batteries, which perform very well. For more information, see my article on The Best Rechargeable Batteries Of 2012.

There are also some lithium-ion rechargeable batteries made for high-performance flashlights and cameras, in these sizes:  18650 and CR123.

A single 18650 battery can replace two CR123A batteries, although at a lower voltage (but much higher amperage). However, the 18650 is a wider cell and will not fit into a flashlight that is designed strictly for the narrower CR123As. Most modern tactical LED lights are designed to use a single 18650 or two CR123As, but it’s best to check before buying. See our article on The Best 18650 Batteries for more information.

There are also numerous other types of lithium-ion batteries made for specific laptops and other electronics gadgets. There is currently no standard size for these lithium-ion cells.

Good books on batteries:

• Batteries in a Portable World: A Handbook on Rechargeable Batteries for Non-Engineers
• Bottled Lightning: Superbatteries, Electric Cars, and the New Lithium Economy
• The Battery: How Portable Power Sparked a Technological Revolution