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.
I thought I'd look at some studies that have tried to answer your question.
The person with the most exposure is Queen Elizabeth II. Her face is on currency, stamps and other everyday items in many countries across the globe. See article here.
Researchers at MIT have created Pantheon. Pantheon curates a catalog of famous people. It's also equipped with search filters. The results of Pantheon (which includes people over all recorded history) suggest that politicians and writers are more likely to be 'famous' than those in other occupations.
If you limit the search to 1970-present, the Pantheon Historical Popularity Index rankings are topped by Eminem.
"Eminem (Singer) was born in Saint Joseph, United States in 1972.
He is featured in 83 different language editions of Wikipedia and has received 108,953,700 combined Pageviews since 2008."
Using the same time period (1970- ), but ranking by Number of Language Editions (a measure of global cultural popularity), it is actor Corbin Bleu who tops the charts.
Corbin Bleu (Actor) was born in Brooklyn, United States in 1989.
He is featured in 193 different language editions of Wikipedia and has received 6,736,571 combined Pageviews since 2008.
While there is no one person who could be defined as the most famous person in the world, there are many people who could be considered. Depending on the interests and activities you might enjoy, you might decide that some of those people are the most famous in the world. Barack Obama, as the president of the United States, is one of the most influential people in the world and is broadcast via the internet, television, radio, and other forms of media 24 hours a day. LeBron James is probably the most recognizable athlete in American professional sports while Bill Gates represents the technological boom that is so important to our global economy.
Queen Elizabeth II is one of the most famous people living throughout Europe, and though she has no political power, the citizens of England honor and revere her and the rest of the royal family.
As one of the worlds main religions, we must consider the leader of Catholicism to be one of the most famous in the world. Pope Francis is the leader of 1.25 billion people in the world. This means that about 15% of our worlds population honors and follows the teaching and philosophy of this man.
Though there is no truly "most famous person in the world" these people are world renowned and are extremely important and influential in our world. Each of these people is known by a large percentage of people in the world and are tremendously famous.
This is not a question that can be answered because it is so subjective in nature. What does it mean to be famous? How can one person be more famous than another?
Due to the ambiguity of the question no one answer will suffice. The reason is simple geography and human nature. I do not watch reality television and would not know a cast member if they slapped me; however, other people know all of them from a variety of shows and can give incredible details about them.
Human interest also plays a role because of influence. Suppose I meet Michael Jordan (consistently on the top of famous people lists) and US President Obama (another top contender) on the street? I know both of them (not personally, but I can recognize them), so who is "more" famous? For me, I would choose Jordan, someone else may want to meet the President.
Geography and culture also play a part. I'm sure in China there are famous people I've never heard about.
If I am required to pick my top three I would go with President Obama, Michael Jordan and Pele (a huge soccer star).
The most famous living person is a matter of debate and is a very subjective choice. We can think of any number of very famous people from different parts of life: movies, sports, Science, technology, business, politics, religion, criminals, celebrities, etc.
The choice will vary based on the respondent's age, geographical location, personal interests, culture, etc. (among a large number of parameters).
Here are few choices, and they are totally personal:
Among celebrities (movie): Steven Spielberg, Tom Cruise, Julia Roberts
Sports: Sachin Tendulkar, Roger Federer, Christiano Ronaldo, Serena Williams
Technology: Mark Zuckerberg, Larry Page, Bill Gates
Politics: Barack Obama, Narendra Modi, Hu Zintao
Science: Stephen Hawking, Craig Venter
Religion: Pope, ...
I can go on, but this is my own list and even this will evolve with time as my interests evolve. So this is a very broad and subjective question and even with a poll of a very wide audience, it will be tough to answer this question.
I agree with @boomersooner, as far as geographical and social factors determining "famous." Whether famous is a quantitative, or based on popular culture. When I read the question, I read famous as universal. So I think of pop icons like Ronald McDonald, although technically not living, or Mark Zuckerberg, or the Kardashians. I know what you're thinking--we should know who Barack Obama is, but a majority of the people range from 15-44 years in the world. This age group is littered with social media platforms and entertainment.