I was thinking about making a TV show about a bunch of teenagers who learn new things everyday from the Stack Exchange sites. Would I need permission to call it "Stackexchange?"
Reviewing https://stackexchange.com/legal/trademark-guidance shows the following:
The logos associated with Stack Exchange Inc. and any Stack Exchange site are a trademark. The purpose of trademark law is to prevent consumers from being misled as to the origin of a product. So if you were making a product, and you used a Stack Exchange name or logo in your product (or in its advertising) in such a way that would mislead someone into thinking that your product was owned by, operated by, endorsed by, or in any way part of Stack Exchange Inc., you would be violating the trademark and this would not be legal.
Our logo images and site names are copyrighted. Any content on the Stack Exchange Network not contributed by users is copyrighted. Copyright is different than trademark. Ordinarily you couldn't copy it. But if you were writing a news story or blog post about a Stack Exchange site, reproducing the logo image would almost certainly be considered "fair use" and allowed under trademark law because you were not using it "in trade."
So the answer to your question is yes, you would need to seek the permission of the trademark holder.