WordPress Self-Hosted Installation
Here are some popular webhosts that offer fantastico:
An alternative host is DreamHost. To quote from their One-Click Installs page:
Fantastico, shmantastico! DreamHost now has its very own one-click software installation!
As a DreamHost customer myself for several websites, I can vouch for the ease of use. It really is one-click!
If you have a dedicated server or a Virtual Private Server (VPS), then the process may be more complex depending on your host. This website is hosted on a 1&1 VPS and the installation is not too different from the one-click processes above.