I've attempted to set up a wiki a couple times but left it go because the complexity overcame the time I had to do it.
Is there a web application that allows you to easily set up a free wiki in 10 minutes the same way wordpress.com allows you to set up a free blog in 10 minutes?
Google Sites (http://sites.google.com) lets you set up a free wiki really quickly. It's a really powerful system with lots of cool features. We use it for our companies internal wiki and it works really well.
The sites you create can be private or public.
Wikia is the most common place I've seen to create community wikis on a given topic.
I like pbworks. Really simple and fast to set up. Some features.
PBwiki is/was exactly what you're looking for. They've rebranded as PBworks and added a host of non-wiki features, but individuals/non-profits can still set up a powerful, functional and free wiki in about 15 seconds.
Here's the signup link: https://plans.pbworks.com/signup/basic20
In theory, all such "set up a wiki in 10 minutes" sites should be listed at the WikiMatrix.
If I were you, I would go through the WikiMatrix Choice Wizard. It asks a bunch of questions. At the "Software or hosted?" question, it looks like you want "I would like my Wiki to be hosted by someone else". After asking all those questions, it gives a list of wiki hosts that meet the criteria you give -- hopefully you can find one that works adequately relatively quickly.
Don't worry about picking the "wrong" host -- if you later decide the host you are using lacks some essential feature, it's pretty easy to pick up all your content and move to a different server, leaving behind a "Moved to ..." message.
