So something like this? The
submerged floating tunnel offers the most immediate path to being able to do this. That, and having untold billions/trillions to spend on infrastructure which if you're going to spend it all on infrastructure, would be far better served on more mundane roads, bridges, etc. in much of rural Africa and Asia.
But you can definitely link all seven continents together using floating tunnels, although right now there isn't much of a reason to put a tunnel between Antarctica and Chile/Argentina. Australia is probably the hardest to do, given the region is prone to cyclones, and also impossible to do without submerged floating tunnels. But you could link Darwin to East Timor, which would be the best link, or maybe Cape York to New Guinea. If either East Timor or New Guinea had been annexed into Australia, I could see more serious proposals for this happening. Of course, you also need to link together all the Indonesian islands plus build a bridge across the Strait of Malacca (more realistically, a tunnel, given the Straits are a major shipping lane).
The attempts to bridge the Darien gap have mostly been thwarted by environmental concerns. If you start earlier, it can be done. The Bering strait is another matter. You wouldn't just need a bridge, you'd need several hundred miles of new highways in Alaska and Siberia. The only way I can see this coming to fruition is if the Russians don't sell Alaska and later decide they need a road connection to their North American territory.
It's thousands of miles of new roads (although Russia is building a road between Magadan and Anadyr, so that's a decent portion of the work already done), and if you want paved roads (built on permafrost), then you need even more. Given the lack of population on either side of the strait and frequently tense relations between the US and USSR/Russia, it's debateable what purpose all this infrastructure might serve. Now, in the case of a "bigger Alaska Purchase", where the Russians at one point seriously considered selling Chukotka alongside Alaska, I could see this definitely being worked on.
An interesting option regarding Russian roads is an additional link between the Bering Bridge and Japan. This would be a road branched off from the Magadan-Anadyr highway, and travel down the Kamchatka Peninsula to connect Petropavlovsk (IIRC the second largest city after Iquitos not reachable by road) to the rest of the world. There you'd have a series of bridges and tunnels connecting the Kuril Islands to each other. The biggest challenges are volcanic eruption in Kamchatka and the Kurils and frequent earthquakes and tsunamis as well. There's some very deep water and long distances betweens some of the islands, so this would have to wait until floating tunnels become more common. But this road/railway would offer a much quicker land connection between the United States, Russia, South Korea, and Japan than the existing roads. In theory, you could also use similar mixtures of floating tunnels and bridges to span the Aleutians-Komandorskiy Islands-Kamchatka route, although the connections there are much longer, but still shorter than an Australia-Indonesia connection.