The Wikipedia of genealogy: a family tree for the entire human race, which everyone can edit and add to.
Although I've often wished such a thing existed, you run into a significant set of problems with accuracy. For example, the Mormons are big on genealogy; many of them are extremely reliable sources, but many are not--e.g., their genealogy files contain "Noah" and similar figures.
On the other hand, if the information is accurate, it raises major privacy concerns. Most reliable sources will not display the backgrounds of living people to the general public. If this were world-write, what's to stop someone from adding my grandparents, parents (thereby giving away my mother's maiden name), and myself without my knowledge? I might not want pieces of this information known--for example, if I were 1/8 $RACE and I lived in an area with a strong prejudice against $RACE, I would not want that fact known. I might not even be aware of this lineage myself--it could come as a real surprise, and possibly an unpleasant one.