Formal:
* Entreating: To plead earnestly for something.
* Supplicating: To ask humbly for something.
* Imploring: To beg someone with great fervor.
* Soliciting: To ask for something in a formal or official way.
* Petitioning: To make a formal request to someone in authority.
Informal:
* Pleading: To ask for something in a desperate or emotional way.
* Cajoling: To persuade someone with flattery or coaxing.
* Wheedling: To persuade someone by using flattery or charm.
* Scrounging: To try to get something for free or by begging.
* Panhandling: To beg for money, especially in a public place.
More negative:
* Squalching: To beg or plead in a servile or abject manner.
* Hounding: To pursue someone relentlessly for something.
* Harassing: To annoy or pester someone repeatedly.
* Shaking down: To force someone to give you something by threatening them.
The best word to use will depend on the specific situation and the tone you want to convey.