Don Vito Corleone already has the corner on fame. (If you don't know who he is, you really should consider another line of work.) Fame is not really what a mob boss is going for; glory and power, yes. But fame will only raise the attention—not to mention the hackles—of law enforcement and other jealous bosses, families, and even your own underlings.
There have been plenty of famous mobsters—and not just Italian ones either. Many countries have known crime families, including Ireland, Columbia, Poland, Russia, Canada, Israel, Japan, Mexico, Puerto Rico, Jamaica, Albania, Serbia, China, and United Kingdom, to name a ton.
Mobsters tend to live fast and die hard (and have lots of entries into crime blotters):
Pablo Escobar (Columbian; cocaine was his business of choice and it made him a wealthy man…who was killed in a gunfight at 44.)
Yaakov Alperson (Israeli; business taker and protection rackets…killed in a car explosion at 53.)
Graham Kinniburgh (Australian; a member of the Moran family, his crimes included murder, escaping legal custody, dishonesty, racketeering, extortion, bribery, possession of firearms, escape, resisting arrest, and assaulting police…murdered at age 60.)
When you're the Boss, it might be wise to be happy with fame in your own family, fame in your own mind, and fame in your mother's mind. John Gotti was famous; he cultivated fame and even reveled in it. And where did it get him (other than six feet underground)?