Wikipedia has published their Transparency Report, including information about the numerous takedown requests they receive. In the report, Wikipedia makes it abundantly clear that they are not in the business of taking down valid information simply because someone wants them to.

Wikipedia received 304 takedown requests from July 2012 to June 2014 — none of which were granted. They also received 54 Digital Millennium Copyright Act notices from that time. DMCA essentially forces Wikipedia to remove "infringing material." Of these requests, Wikipedia only found 28 to be valid. They made these removals abundantly transparent and evident. In both cases, the vast majority of requests came from the United States. 

Here are some of the more notable takedown requests Wikipedia rejected:

This Monkey Selfie:

via Wikipedia

A photographer set down his camera in Indonesia, and a curious macaque monkey took the opportunity to take this fantastic selfie. She also took some other photos, but this particular selfie is really top notch. It made the rounds on various publications and was added to Wikipedia Commons. 

The photographer requested it be taken down, claiming copyright infringement. Wikipedia disagreed and kept the photo up.

This Aboriginal Language:

The Tasmanian aboriginal language center claimed to own 'palawa kani,' which Wikipedia describes as "a constructed language" and "a project to create a generic language resembling the extinct languages once spoken by Aboriginal Tasmanians." Wikipedia did not remove it because no one can claim ownership of an entire language, even a constructed one. 

This French Intelligence Agency:

The Direction Centrale du Renseignement Intérieur took Wikipedia editing to the next level when they brought a Wiki administrator into their (probably really intimidating) offices. The admin was threatened with criminal penalties if he did not delete information about a military base the agency believed was classified. As it turns out, the base was not classified at all, and had even offered tours to the press.

These Classic Books:

A publishing company forgot about public domain when they requested that Wikipedia remove Pride and Prejudice, Persuasion, The Adventures of Sherlock Holmes, and Fables from their Wikisource translations, all which are out of copyright.

Babe Ruth's Called Shot:

via Wikipedia.

This image shows Babe Ruth's famous called shot, in which he hit a homerun to in the 1932 World Series after allegedly predicting where it would go. There has been much debate as to whether he actually pointed in that direction or not, and this still from a 1932 film arguably shows Ruth did point. The owner of the film believed he had copyright of the still, sending a DMCA takedown request to Wikipedia. However, Wikipedia did not remove the image, citing fair use.