Today I played around a bit with the stuff done by the Wii homebrew/hacking community, which is far more advanced than I had expected.
I had the game Twilight Princess borrowed by a friend. For those who don't know, using this game you can run homebrew software on the Wii without the need of a modchip.
The way this works is that first you install a modified savegame for Twilight Princess, afterwards you can install the Homebrew Channel. The Homebrew Channel will stay in the main Wii Menu and you'll be able to run various stuff from there. While looking around the various webpages covering the topic, I found that
wiibrew.org is a good place to find documentation and links to the appropriate projects. So look there for the details.
The gc-linux project (originally porting linux to the gamecube) has a simple Linux image available.
A warning: Nintendo released a new firmware that stops this method to work, so if you wanna have fun, don't update your Wii. And the obvious warning: Everything you do is at your own risk.
Update: Seems the Firmware update
is no longer a problem.