Fame is an fascinating topic. Looking at the posts here, fame is also clearly something of a hot-button issue.
In a cultural moment that stresses the value of recognizing cultural bias, it is no wonder that a discussion of fame turns to considerations of relativity and to attempts to qualify the question. Such considerations and qualifications are entirely appropriate. They do not, however, diminish the interest of the question. They actually enhance the interest of the question.
When we ask who is the most famous person living today, we are also asking, in part, about what cultures are best represented in the world.
In a study conducted over the past few years by a European organization, Michael Jackson, Lady Gaga and Justin Bieber were found to be the most famous people in the world based on their social media connections. Michael Jackson was not alive at the time of the study (2013), which points to another quirk of this question.
When we discuss famous figures who are alive currently, we are side-stepping the fact that this person's fame will undoubtedly rely much more on (impersonal) repeated media images than it will on face-to-face (personal) interactions. Thus fame is best seen as a product of something larger than any single person. Fame, when addressed on the scale of "most famous" is best seen as a collaborative result of the workings of a vast network of advertisements, conversations, news stories, etc.
As much as a study of social media connections may try to reduce cultural bias in its assessment of fame, there are some points we should make there too. Some countries may have access to social media at greater or lesser rates than others and so judging fame by social media connections continues to skew the discussion of fame toward certain cultures that embrace certain technologies.
One interesting note here is, again, the idea that the list of famous people in the study cited here is made up of two Americans and one Canadian. Barack Obama is mentioned in discussions of "most famous living people" as well and we can see a trend in this set of names. The most famous people in the world are figures from the cultures that are most widely represented in the world via media of all sorts.
The same argument applies when considering where fame/celebrity seems to be drawn from on the scale we are discussing. The most famous living athlete is probably Christiano Rinaldo (who plays the world's most widely watched sport). But more people listen to music than watch sports, it would seem, so the world's most famous musicians are more famous than the world's most famous athletes. The individuals concerned here can be seen as representatives of their respective sub-cultures.
As other posts have pointed out, there is no good data available to determine a definite answer to the question of who is the most famous person living today. My guess is that the most well-known politicians are Barack Obama, Vladimir Putin, Angela Merkel and Kim Jong-Un (each famous for different political and cultural reasons).
My best guess, to offer a single answer, is that the most famous person living today is Barack Obama. There is a likelihood that more people would recognize his name and image than any other living person.