How common is it for large bridges or bridge-tunnel complexes to not have toll?

For a North American answer, the Pont Champlain (Champlain Bridge) in Montréal (which is 3.4 km long, compared with the Golden Gate's 2.7 km, so "large" by this question's metric) currently has no toll.

As in other systems, a toll was enacted to pay for its construction and maintanance (1967), which was later removed (1990).

This bridge also welcomes 159 000 cars daily, nearly 40 000 more than the Golden Gate Bridge.

Very few bridges or tunnels in the UK have a toll.

The Queensferry Crossing near Edinburgh in Scotland has no toll despite opening very recently, at the end of 2017. As it is almost exactly the same length as the Golden Gate Bridge (2700m versus 2737m) it seems like a good example.

The Severen Bridge Crossing in the UK is currently a Tolled bridge but increasingly looks like the Toll will be removed. It recently went from Private to Public ownership as per the terms of the contract when it was built and Government have stated the Toll will be removed.

In many cases the existance of a toll is dependant on the terms under which the Tunnel/Bridge was built. If the funds are not available to pay for building it straight away then either the company will be given a lease to allow them to charge for use for a period of time, during which time they will make back the costs of building and maintaining as well as a healthy profit. Where it is decided that the bridge or tunnell should be free to use then Government must pay the full cost of building.

The Zeeland Bridge in the Dutch province of Zeeland was opened on 15 December 1965. It is a little over 5 kilometres long. Toll was levied from 1965 to 1992, to pay off the construction of the bridge and to set up a maintenance fund. It was made toll-free on 1 January 1993.

Likewise, the Western Scheldt Tunnel in Zeeland was opened on 14 March 2003. It is 6.6 kilometres long. Toll will be levied until 2033, although from time to time people argue it should be made toll-free earlier.

The newly-built Crimean Bridge is toll-free, which of course is slightly political.

According to the Bureau of Transportation statistics, there are 140 toll bridges in the USA.

The US department of transportation - Highway administration classifies 57627 bridges as "Principal Arterial - Interstate" (25231 urban and 32396 rural), out of more than 600 000 bridges total. So, only 0.25% of the bridges which the feds define as "principal" are toll bridges.

Since I don't know the criteria for defining a bridge as "principal", I also looked up some data by length, and found that there is an interactive site where one can search national bridge inventory data from the USA. If I place the length cutoff at 5280 feet (a mile), I get 296 results. If I decide that a kilometer-long bridge (3281) is a "large bridge", I get the message "Your search returned more than 500 results. You may want to go back and narrow down your search choices." and only the first 500 results are shown.

So, the answer: even if we restrict the country to the USA (where the OP built his intuition) and the definition of a large bridge to "longer than 1 mile", toll bridges aren't the norm in the sense that they are less than 50% of all bridges. Relaxing these conditions will yield even more tollless bridges, I don't think there is a need to enumerate them even if they are very prominent.

Germany has a general toll for all commercial vehicles heavier than 3,5 metric tons on all controlled-access highways and some federal highways. As far as I am aware, there is no instance of any sort of toll in addition to that, in particular, I am not aware of any toll that applies only to a specific section of road, such as a bridge or tunnel. The truck toll is based solely on distance driven (but not "where", only "how far"), number of axles, and the "Schadstoffklasse" (roughly "pollution class") of the motor.

There are privately operated stretches of highway in Germany, but those are paid for by other means (usually, the government, or the operator receives a fixed percentage of the above-mentioned truck toll). The operators don't collect additional tolls themselves.

There may be tolls on ferries, though, but many are "free" (meaning, they have a contract with the municipal or state government(s) on both sides of the river crossing).

Switzerland has a single flat-fee yearly toll, but I have never encountered location-specific tolls, not even at major arteries such as the Gotthard tunnel.

For me, as a German, the idea of a location-specific toll seems very strange. Roads are a vital part of infrastructure, and I expect them to be provided by the government. That's what I pay my taxes for.

The truck toll is a special case:

• Heavy commercial trucks cause a disproportionate amount of wear on the roads
• At the same time, they earn money from those roads, and even more money from well-maintained roads
• Germany is a transit country between the North Sea / Baltic Sea and the Mediterranean Sea, meaning that a disproportionate amount of trucks are not registered, owned, or operated in Germany, thus paying no taxes

Yes, there are large bridges that don't have tolls. This is true in countries around the world, but given that you're specifically asking about the US, one example is the Seven Mile Bridge, which as the name implies is around seven miles or 11 kilometres long (and thus meets your "very large bridge" criteria), and has no toll. The dozens of other bridges around this area of the Florida Keys also have no tolls.

For tunnels, the longest road tunnel in the world is the Lærdal Tunnel in Norway, which is over 15 miles or 24 kilometres long and has no toll.

These are just two, but there are many more both inside and outside of the US, just as there are many that do have tolls.