What are some historical examples of laissez-faire capitalism? I suspect both supporters and detractors of laissez-faire capitalism have some historical examples in mind that they consider reasonably close to an ideal or to a "natural experiment". (The examples used by supporters vs detractors might not actually be the same though.)

The Wikipedia article I've linked to discusses at length the history of the concept itself, but I'm having trouble finding a discussion of examples of the realization of the concept in there.

So, do these examples exist in literature? If so, what is the pro/con angle on them?
There is an ongoing project to try and estimate how close a society is to the ideal, see the indices of economic freedom by the Heritage Foundation and the Fraser Institute. The ones who made They score countries from 0 (all-pervasive state control) to 10. Hong Kong (8.91) and Singapore (8.71) are currently the two countries closest to the ideal, according to the Fraser Institute. 

These projects only look at societies from 1970 and onward, however.

A few western countries during parts of the 19th Century have, in a way, been relatively close to the ideal, such as the US up till the Progressive Era and Sweden (around 1860 to 1910). However, both societies restricted the rights of married women and the southern states upheld slavery and later on Jim Crow laws against blacks. Still, the economy was mostly coordinated through private effort, though there were legal barriers for women and blacks to get into the "commanding heights" of these economies.

For a description of the decline of laissez-faire in the US, see Murray Rothbard's The Progressive Era and Robert Higgs' Crisis and Leviathan, which should complement a general economic history of the era.

Mancur Olson, an economist and critic of laissez-faire, argued in The Rise and Decline of Nations that (probably the last) 50 years of British India was among the closest to a laissez-faire ideal. During that time India had hardly started developing. He says that the caste system was a drag on the economy, but he didn't see castes as being in conflict with the freedom to enter any business or trade.

Another way to study laissez-faire systems is to look at specific markets, when not heavily regulated. The Voluntary City shows examples of the private production of urban infrastructure, roads, commercial courts, police, social insurance (healthcare, sick benefits, life insurance), and primary education in 19th century US and England. For examples of the private production of coins, see Good Money; for bank notes (and a free banking sector) see The Theory of Free Banking.

Also, for the record, I'm an ancap.