I think it takes a special kind of audacity to be one of those celebrities that insist on only being called by one name.  You know, like Cher or Drake or whatever.  I mean, more power to you if you can pull it off I guess.  When someone says Cher I know exactly who they're talking about.  Still, it shows a lot of confidence/arrogance to think I'm so great everyone will know me just by my first name.  I guess sometimes it kind of happens on its own though, too.  As far as I know Elvis Presley wasn't pushing to be known by one name, but when you say Elvis, that's who comes to mind.

I think it only works if you have a sort of uncommon name.  I think anyone named, say, John is probably out of luck in terms of only being known by their first name.  On the other hand when you say Shaq I for sure know who that is, and it's not even his full first name!