Informal:
* Barking
* Chattering
* Gabbing
* Blabbering
* Nagging
* Whining
* Complaining
* Grumbling
* Squealing
* Shrieking
* Pestering
* Bickering
More formal:
* Talking incessantly
* Speaking rapidly
* Making a lot of noise
* Being voluble
* Being loquacious
The best synonym will depend on the specific situation and what kind of tone you want to convey.