http://en.wikipedia.org/wiki/BgC3
has anyone ever heard of anything related to this company? I have heard a number of theories, mainly concerning it being a patent troll.
BGC3 appears to be Bill's business presence now that he's not at MS on a day-to-day basis.
A very common business design pattern for wealthy businesspeople is to have a "parent company" that is essentially the businessperson plus their entourage: the legal team, technical due diligence buddies, capital raising marketing staff (people who woo wall street). This company may or may not be owned by a parent company that does the investing in other companies that actually "do stuff".
Without seeing the evidence for the patent troll concerns, I have no reason to believe it's the case -- he just needs some place to park his Mercedes during the day. :)