Is there a way of measuring fame, as who is the most famous?

Here is one quantitative approach:

"Our method of celebrity classification relies on the fact that human responses to stimuli are not linear. For example, a first magnitude star is 2.5 times brighter than a second magnitude star, which is 2.5 times brighter than a third magnitude star, and so on (Pogson 1856).

Such a relationship is called logarithmic. Many scientists since the late 19th century have believed that the responses of our senses of sight, hearing, smell, taste, and touch are logarithmic (e.g., Fechner 1860).

In this paper we propose that the Weber-Fechner Law of human perception also applies to fame, such that people we perceive as 'A' List Celebrities are on average ten times more famous than people we perceive as 'B' List

Celebrities, who are on average ten times more famous than people we perceive as 'C' List Celebrities, and so on.

We classified people in seven different fields (business, film, music, politics, religion, science, and sports) as a function of their fame in terms of the logarithmic international standard unit of fame, the dBLw..."

Read on here:

https://en.wikipedia.org/wiki/A-list

http://www.biographyonline.net/people/famous-100.html

Newspaper column inches was one way of gauging fame.
We also used to hear about the most photographed celebrities, such as Marilyn Munroe and Princess Diana. Exactly how fame is measured is beyond me. A celebrity from the U.S.A or UK could go un-noticed in Ukraine. World-wide recognition from a photograph will be few and far between.

The amount of people that can recognise your image is a good way to measure fame, and the most famous are those that are recognisable internationally.

This woman's fame is hard to beat -

http://www.telegraph.co.uk/news/uknews/1400202/The-most-famous-face-in-the-world.html

There are lots of ways:

Polls
Times mentioned in print
Times on magazine covers
Times on coins or stamps
Number of cult followers
Etc.

The Gallup Poll is more scientific, but reflects only what living Americans tell them.

http://www.gallup.com/poll/1678/most-admired-man-woman.aspx

Another way to measure fame is by the number of Google hits found when a name is searched for.

http://www.newsfactor.com/story.xhtml?story_id=24133