Last year I discovered that I (and of course other members of my immediate family) are descended from Alexios I Comnenos, one of the greatest of Byzantine emperors, who reigned from 1081 until his death in 1118. He is variously my 25th through 29th great grandfather.
How unusual is it to have aristocratic and royal ancestors? Not at all. If we go back 20 generations, multiplying each successive generation by 2, each of us supposedly has 2,097,152 ancestors of that generation alone. After 30 generations each of us apparently has 2,147,483,648 ancestors of that generation. After 40 generations we each have 2,199,023,255,552. These numbers are clearly fantastic, because the numbers of human beings that have ever lived on earth are far, far short of 2 trillion! This means that if we go far enough back we are descended from the same fairly small number of people many times over, particularly if our origins are in the same part of the world.
For example, each of us in my line of the family is descended from Charlemagne in more than 100 different ways through at least three of his offspring. Thus anyone claiming special status for being descended from royalty is making an exceedingly insignificant claim. It certainly will not put one in line for any thrones! The real pleasure in researching genealogy comes to those already possessing a love of history in general.
Coincidentally, shortly after I came to the above conclusions, confirmation appeared in an article by Steve Olson, "The Royal We", The Atlantic Monthly, vol. 289, no. 5 (May 2002), pp. 62-64. Citing Mark Humphrys, who maintains a website titled "Royal Descents of Famous People", Olson argues that "everyone in the world is descended from Nefertiti and Confucius, and everyone of European ancestry is descended from Muhammad and Charlemagne."
Before the fall of his regime, Saddam Hussein had employed professional genealogists to demonstrate his descent from the prophet Muhammad, obviously to enhance his legitimacy among traditional Muslims. As it turns out, virtually any westerner he would chance to meet (not to mention his fellow Arabs) is also descended from Muhammad. It's simply not a statistically significant claim.