What stands out for me is that of the novel Nightfall by Issac Asimov and Robert Silverberg. The fact of constant daylight underpins everything else in the story.
The fictional planet Lagash (Kalgash in the novel adaptation) is located in a stellar system containing six stars (Alpha, Beta, Gamma, Delta are the only ones named in the short story; Onos, Dovim, Trey, Patru, Tano, and Sitha in the novel), which keep the whole planet continuously illuminated; total darkness is unknown, and as a result so are stars outside the solar system. A group of scientists from Saro University begin to make a series of related discoveries: Sheerin 501, a psychologist, researches the effects of prolonged exposure to darkness, Siferra 89, an archaeologist, finds evidence of multiple cyclical collapses of civilization, and Beenay 25 is an astronomer who discovered irregularities in the orbit of Lagash around its primary sun Onos. Beenay takes his findings to his superior at the university, Athor, who formulated the Theory of Universal Gravitation. This forces the astronomers at Saro University to attempt to find an answer to what is causing this anomaly. Eventually it is discovered that the only thing that could be causing the deviation is an astronomical body that orbits Lagash.
Beenay, through his friend Theremon 762 (a reporter), has learned some of the beliefs of the group known as the Cult ("Apostles of Flame" in the novel). They believe the world would be destroyed in a darkness with the appearance of Stars that unleash a torrent of fire. Beenay combines what he has learned about the repetitive collapses at the digsite, and the new theory with the potential of eclipses and concludes that once every 2049 years the one sun visible is eclipsed, resulting in a brief 'night'.
Since the population of Lagash has never experienced universal darkness, the scientists conclude that the darkness itself would traumatize the people and that the inhabitants of the planet would need to prepare accordingly. When nightfall occurs, however, the scientists—who have prepared themselves for darkness—and the rest of the planet are most surprised by the sight of previously-invisible stars outside the six-star system filling the sky. The short story did not cover what happens after that, but in the novel and X Minus 1 program, civil disorder breaks out; cities are destroyed in massive fires and civilization—as previously known—collapses, with the ashes of the fallen civilization and the competing groups trying to seize control.
http://en.wikipedia.org/wiki/Nightfall_%28Asimov_short_story%29