First of all, the U.S. only allows 11 of those 13 channels. Additionally, the original wifi developers made a mistake, of sorts, and signals within channels bleed over to their neighbors...
there are really only 3 channels you should use: 1, 6 and 11.
That said, you can have far more than 3 devices on wifi at a time, because devices will share time on each channel. It's just like having someone listen to several conversations taking place at once in a crowded room: not everyone is talking all the time. If two people talk at the same time, the listener may have to ask one or both to repeat themselves. The more people you add to the room, the less total information you can pass around, because people will constantly interrupt one another at an increasing rate. A good rule of thumb is around 25 devices per channel for casual browsing, but this can drop significantly for non-casual traffic like gaming, p2p file sharing, video streaming, and large file transfers.
In networking parlance, we say a wifi cell is unswitched and half-duplex, making it very sensitive to collisions. Wired networks typcially don't have these weaknesses (switched and full-duplex), and are also much less susceptible to random electromagnetic interference. While wifi is a "good enough" technology to use at home, serious networks or any serious application will do much better pushing as many people or as much network traffic as possible to a wired connection.
I run the campus network at a small college, and it's sad to see how many new students arrive this year who have never used a wire for network access. They think the notion of needing a wire is quaint, and don't understand the physical limitations involved, and why 80 devices (nearly 2 per student on average) in dorm space the size of their parents' house doesn't work so well. Re-educating them about this is hard.