More formal:
* Beseech
* Entreat
* Supplicate
* Plead
* Solicit
More urgent:
* Beg
* Cajole
* Importune
* Press
* Urge
More desperate:
* Cry out
* Appeal
* Call upon
* Vouchsafe
More specific to a particular subject:
* Exhort (to urge strongly)
* Adjure (to command solemnly)
* Conjure (to call upon earnestly)
* Invoke (to call upon for help)
The best synonym for "implore" will depend on the context of your sentence and the specific emotion you want to convey.