Formal:
* Implore: To beg urgently or piteously.
* Entreat: To ask earnestly or urgently.
* Solicit: To ask for something earnestly.
* Supplicate: To ask humbly or submissively.
* Beseech: To beg earnestly.
Informal:
* Beg: To ask for something earnestly and humbly.
* Plead: To ask for something in a way that shows you are desperate.
* implore: To beg someone earnestly and desperately.
* Cajole: To persuade someone by flattery or gentle urging.
* Wheedle: To persuade someone by using flattery or trickery.
* Appeal: To make a serious or urgent request for help.
* Yearn: To long for something desperately.
* Crave: To have a strong desire for something.
Other:
* Petition: To make a formal request.
* Request: To ask politely for something.
* Demand: To ask for something in a forceful way.
The best synonym to use will depend on the specific context and the desired level of formality.