Formal:
* Implore: To beg someone earnestly or desperately to do something.
* Entreat: To ask someone earnestly or formally to do something.
* Supplicate: To ask humbly for something, especially from a god.
* Solicit: To ask someone for something in a formal or official way.
* Petition: To make a formal request for something.
Informal:
* Beg: To ask someone earnestly for something.
* Plead: To ask someone for something in a desperate or emotional way.
* Appeal: To make a strong request to someone.
* Importune: To ask someone persistently for something.
The best synonym to use will depend on the specific context and the tone you want to convey.