Yes, there are famous exorcists. Perhaps you've heard of some of these people, perhaps not:
Father Gabriele Amorth, who has apparently performed tens of thousands of exorcisms, is one of the founders of the International Association of Exorcists (IAE).
Father Jeremy Davies, who believes that homosexuality, acupuncture, yoga, and Harry Potter are "caused" by a "contagious demonic factor" was also one of the founders of IAE.
And let's not forget Father Damien Karras, a fictional priest (from the movie The Exorcist), but somehow a real person in many moviegoers' minds.
But if it's fame you're seeking in your profession, becoming an exorcist probably isn't the most efficient way to do that. Honestly, if you're so famous that the devil already knows who you are before you even show up to attempt to destroy him, he's going to be aware of how you work and will always be one step ahead of you.