The Wikipedia page has since been rephrased to contain the following, more accurate, paragraph:
The maize plant is often 3 m (10 ft) in height, though some natural strains can grow 13 m (43 ft). The stem is commonly composed of 20 internodes of 18 cm (7.1 in) length. A leaf, which grows from each node, is generally 9 cm (4 in) in width and 120 cm (4 ft) in length.
What this means is that occasionally, a natural strain of maize can grow to be 13 m (43 ft). The original paragraph "While natural maize varieties grow to 12 metres (39 ft) tall..." makes it seem like most natural maize varieties grow this tall. This is not the case.
The Guinness World Records has a record for the "Tallest sweetcorn (maize) plant."
The tallest sweetcorn (maize) plant measures 10.74 m (35 ft 3 inches) and was grown by Jason Karl of Allegany, NY, USA. It was measured on 22 December 2011. The Maize was measured in the horizontal position from the bottom of the botanical shoot to the top of the tassel.
The measurement of 10.74 m was in 2011. On April 1, 2017, The Scientist reported that the plant was 45 feet tall. The article very nicely describes the process in non-technical terms (emphasis added).
At his family’s farm, he grew the Chiapas 234 variety up to 35 feet tall, and that was just by manipulating environmental variables such as night length. That plant earned him the world record for tallest corn plant in 2011. Then, to push the plant taller yet, he bred a Chiapas corn plant with a mutant plant carrying the Leafy mutation and then back-bred that hybrid to the Chiapas for six generations to essentially place the Leafy mutation in the Chiapas genetic background.
But before he grew his new corn variety, there was one last environmental variable to consider: the growing season. Karl could only grow for about seven months in New York before the costs of heating the greenhouse became prohibitive. So last year, on his own dime, Karl moved from Olean to the central valley of Costa Rica. “In New York, you have to keep corn from freezing, whereas down here you can focus on trying to grow it out to completion to see what’s happening,” he says. Late last year, his efforts paid off. In a makeshift greenhouse setup designed to both encourage the plant’s growth and support its stem as it climbed toward the sun, he grew a corn plant that measured 45 feet tall with more than 80 internodes—56 more internodes than unmodified Chiapas corn grown under normal conditions. (He regularly publishes his data in the Maize Newsletter.)
Farm Show Magazine contains a picture of the greenhouse ("55-ft. tall") with the corn inside as well as the following paragraph.
[The corn breeder's] tallest plant so far is the result of 7 generations of breeding a tropical strain called Chiapas 234 after crossing it with a naturally occurring mutation that adds internodes. Normally Chiapas 234 has 18 internodes below the ear and 6 above. The more internodes and elongation between them, the more leaves and the taller the plant.
I do not see any reason to doubt the sources as your main concern is "The source seemed to be a dead end that required a password" and now there are new sources (available after you posted your question).
Does "natural" corn grow really tall, or is Wikipedia wrong here?
The answer to this question, of course, depends on the definition of "natural." The tall corn plant is the result of selective breeding of an already tall plant species. So, does "natural" include human intervention? If yes, then a natural corn plant really is 45 feet tall. If "natural" means "in the wild," then no corn plant is natural as the modern-day maize was domesticated with selective breeding. Yes, there is a selectively bred maize plant that is 45 feet tall. The maize plant is somewhere in Costa Rica (see sources).
This question, while dating back to 2012, is still relevant as the Wikipedia claim is still there (albeit in a modified and even more striking form) and it doesn't appear likely that we will stop planting corn anytime soon.