If you go back to the Middle Ages, Obama has ancestors in most of Catholic (and Protestant as they were Catholic in the Middle Ages) Europe. Once you find one ancestor in the nobility, the tree quickly become international.
Apparently some genealogists calculated that a third of all Americans may be descended from King John (of Robin Hood fame), and through him a big chunk of the English, Scottish, French, Norman/Viking, Belgian (e.g. Flanders, Brabant, Hainaut) and German nobility of earlier centuries. Virtually all British people have royal ancestors and therefore traceable ancestors in historical times many European countries, including Poland. There is nothing special about this.
Anyway, so many generations elapsed that, even if the paper trail is correct, there may be no inherited DNA due to recombinaisons at each generations cutting out part of the ancestry in favour of others. As you know it is wrong to say that we inherited 12.5% of our DNA from each great-grand parent. It might be 9% or 15%. It's sad that they still teach that in most universities though (even at MIT from what I saw on their online courses). It's simpler to teach it this way, by simplicity is not a virtue when it's wrong. If only people learned that after 7 generations, you do not inevitably inherited a mathematical 0.78125% of each ancestor's genome but that some ancestors contribute to more than 1% and others to 0%. And that's barely after 7 generations. There are 38 generations separating Obama from Mieszko I. In other words, even if his genetic contribution was not eliminated by recombinaisons somewhere along those 38 generations, the amount contributed would be less than 0.0000000001% of the genome. Any European would almost certainly share an incredibly greater amount of DNA (many tens of thousands times greater) with a random Chinese, Indian or Ethiopian than that figure.
All this to say that single ancestors on a family tree do not contribute anything meaningful to one's genome after a few generations. The only ancestor that keeps contributing the same 1% of DNA generation after generation, even after 1000 or 2000 years if the one on the patrilineal line for men. I wonder why people still care about genealogy. It's almost as meaningless as astrology.
Genetic inheritance is much more complex than most people realise. Back in 2009 I posted an article explaining, among others, how men (but not women) inherit from DNA from their mothers, and why some children were closer to one parent genetically (and also in terms of looks, character, tastes, etc.). It's not something that people understand instinctively, and indeed I have met a lot of people who have been to Med school or studied biology and had never thought about it.