There are math experts who assert that everyone is a cousin of every one else, mainly because there had to be a Most Recent Common Ancestor (MRCA) - someone who "fathered" or "mothered" the species as we know it. See this article for further discussion.
Jack Lee makes an interesting analysis that shows that the odds are excellent that every person with European ancestry has Charlemagne in their ancestry. His point is:
My conclusion, which was surprising (to me at least), is that there is virtually no chance that anyone of European ancestry is not directly descended from Charlemagne.
Here's my reasoning. Charlemagne was approximately 40 generations back from the present day. Each person has 2 parents, 2^2 = 4 grandparents, 2^3 = 8 great-grandparents, ... and 2^40, or approximately 1,000,000,000,000 (one trillion) 40th-generation ancestors, which means half a trillion male ancestors. Of course, since the entire male population of Europe at the time of Charlemagne was only about 15 million, these half trillion ancestors cannot all have been different men -- obviously there has been a lot of cross-breeding, and many of our ancestral lines cross and re-cross, eventually ending up at the same person.
He then uses probability theory to show that the odds that Charlemagne is NOT one of your half-trillion male ancestors is an incredibly small number: about one chance in 10^15,000 (thats 10, followed by 15,000 zeroes). Read the whole article.
Our challenge as genealogists, of course, is to find a line back to the "giants of history" - Charlemagne, Alfred the Great, William the Conqueror, Niall of the Nine Hostages, etc.
Have you found the link yet? If not, keep searching. There are several books out about descendants of European royalty and nobles, and there is a thriving interest in medieval genealogy. If you can find a link to one of the persons mentioned in these books, you're in luck.
You can legitimately walk up to Buckingham Palace and ask for dear old cousin Elizabeth, asserting your relationship. She would probably send out her lowest footman to see you, since you are certainly his cousin too. And, very likely, the homeless person in the park downtown is too.