In a discussion among a group of friends we considered the positive values and disadvantages of being famous. Stewart Brand argued that fame generated an ease of entry into a world of influencial people and access that allowed many unique options for creating an interesting life.
It has been thirty five years since that discussion and I am now able to look at two of the disadvantages that we didn' t discuss at the time.
One is that fame can associate you with a failed enterprise, whether you totally supported it or not or whether you made other contributions. 
Example: Arthur Schlesinger Jr. Schlesinger Jr. was a prominent historian, closely associated with the Kennedy clan and the New Deal. The New Deal is the failed version of American socialism with high taxes, expansive government, labor unrest and economic feebleness. Not only did New Deal ideals fail and bring Schleshinger Jr. down, but the Kennedys have earned a reputation as philandering playboys. Not good for Schleshinger's reputation either.
Another negative is that fame can give you access to a social firmamentarium of people who think and talk alike and make sure that you are hip, current and 'with it' but no longer original, if you were original to start with.
Two strong arguments against fame, from my perspective.