I know this is a hotly debated topic among those with claim to the title and I'm admittedly a big time homer for this angle, but golf was first played in North America in Charleston, SC in 1739 on Harleston Green on the downtown peninsula. While this predates the formalization of the rules of golf by the R&A and the acceptance of both the 12 and 18 hole standards as well as the work of Old Tom Morris, I do wish someone would come across some record or account of a routing that was followed by the Harleston Green Golf Club. The club itself lasted until the early 1800s, losing out on its playing grounds to a military installation in the War of 1812 and subsequent logging ponds that were later filled to become some of our most architecturally significant neighborhoods.
I'd be curious to know what the oldest account of, or actual routing plan exists for any course - I assume there was some writing by early players eluding to their preferred directions of play towards the Eden estuary that may give some hint to how much evolution has occurred at the Old Course, but it seems that golf, pre OTM is largely prehistoric in nature.