The Cape to Cape trip: drive the same vehicle from the Cape of Good Hope to Cape Horn. Yes, this is theoretically possible (if you combine a few recent world records). It has yet to be done in a continuous basis, however. You could be the first, a modern day explorer! This is, without a doubt, the ultimate driving adventure in the world.
There are three main challenges to this, aside from endurance. First is managing to cross the Bering Strait in a car. Do you make your vehicle amphibious, or do you try and drive across the ice during the 3 week interval in the winter? Someway you need to get across the over 50 miles of water that separates Russia from Alaska.
The second challenge is getting across Siberia after Yakutsk or so, when the roads sort of peter out. Or become trails, or ice roads. The main route seems to be somehow getting to Pevek, and taking the ice road to Uelen. And once you get to Alaska, it isn't exactly a picnic to get to Fairbanks, where real, continuous roads start. The third challenge is crossing the Darien Gap in Panama, with its lack of road through the rainforest. But after CROSSING THE BERING SEA IN A CAR, these seem trivial.
So far Steve Burgess and Dan Evans of Wales have come the furthest. They started in the UK, drove to Yakutsk, and took an ice path along the top of Siberia to Uelen in their highly modified Land Rover. Putting on pontoons they had carried on the roof, they successfully made it across the Bering Sea: the Rover became a boat, with the engine turning an outboard propeller. Upon arrival in Wales, Burgess and his team them made it to Nome. However, throughout this trip, there has been lots of back and forth flying, support vehicles, and the trip is currently on hold as he raises money to get from Nome to Cape Horn.
In short, it can be done, but has not been done yet. People occasionally make the trek across the Bering when it is frozen via snowmobile to see family in Russia or the US, whichever the case may be. A man and his son have skiied across.
Now if only you could get some sort of highly modified motorcycle, maybe with a sidecar, small enough to get around the bush, light enough to be able to get across the ice without falling in.... that would be a journey. Ewan McGregor stopped in Magadan and took an airplane to Anchorage. That's cheating. No airplanes, no boats, just one land vehicle, Cape Town to Ushuaia. Enough said.