Short Answer: I recommend Anvow Ni-MH rechangeable AAAA batteries.
Yes, rechargeable AAAA batteries do exist.
AAAA batteries (or quadruple As) are most often used in small devices such as laser pointers, LED penlights and small headphone amplifiers.
AAAA batteries are not common. However, rechargeable AAAAs do exist in two forms: NiCads (Nickel Cadmium) rechargeables and NiMH (Nickel-Metal Hydride) rechargeables.
NiMH are preferable — they have a higher capacity and are less toxic than NiCads (cadmium is a potent environment toxin).
Here are all the commonly available rechargeable AAAA available: