Knowing how important genealogy research is to LDS leads me to wonder how far back any of us could trace our family tree. Have any LDS members been able to trace back to any biblical characters?
Robert C. Gunderson, Senior Royalty Research Specialist, of the Church Genealogical Department stated, in regards to the possibility of accurately tracing back to Adam and Eve, that the answer is No. The reason being that European royal genealogy before 500s A.D. Cannot be verified. This would exclude biblical characters as well.
It's probably not possible/easy to know who has traced the furthest back since there are so many people doing it and their records aren't all public.
Between neglect, wars, natural disasters, and other catastrophic events destroying important records throughout the ages, and people not bothering to keep them in the first place, it's next to impossible to find genuine family history data more than a few centuries back... which doesn't mean people haven't tried, both in our own time and in ages past!
You will occasionally hear someone talking about having a line that goes all the way back to Adam. What this means in practice is that they managed to trace back far enough to link up with a royal genealogy of some king back in the day who had their family line connected to some important Biblical lineage for political reasons. Such royal genealogies are, of course, considered highly unreliable; it's just that no one really knows for sure where the accurate records end and the mythologizing and politicizing begins.