I was up there this June. There are two Business 3 signs, both on Trunk Road near its interchange with the Parks Highway freeway (state route 3). They indicate that Business 3 follows the frontage roads on each side of the freeway north from Trunk Road. However, there is no indication of where Business 3 ends on each side of the freeway, such as reassurance or "end" shields on the frontage roads, or junction signs at the next interchanges to the north.
Alaska DOT&PF's route logs don't indicate the extent of Business 3 (nor do they delineate where other numbered routes begin and end -- the logs are organized by highway/road name, the only route numbers are internal six-digit inventory numbers). I don't recall either of the frontage roads (each a named road) showing up in the route logs, which include all roads maintained by Alaska DOT&PF and many that aren't.
I looked at the back of one of the Business 3 shields, suspecting that the signs might be unofficial. But it had an Alaska DOT&PF sticker, indicating the sign was erected in 2004.
Business 3 remains a mystery. And I haven't seen any other signed business routes elsewhere in Alaska where one might expect them if there were any, such as along the old Richardson Highway through downtown Copper Center, or the fragments of the old Glenn Highway through Palmer and Eagle River.