Would it be technically possible to program the arduino on the arduino?
As cool as that would be: no, it's not feasible.
The Arduino does not have the memory or horsepower to run a C compiler.
What about TCC : Tiny C Compiler ?
(I'm not even sure if Arduino is x86 by the way)
(I'm not even sure if Arduino is x86 by the way)
It is not!
SMALL! You can compile and execute C code everywhere, for example on rescue disks (about 100KB for x86 TCC executable, including C preprocessor, C compiler, assembler and linker).
You have 2K of RAM on an arduino.
Oh, I see.. thanks
Maybe on DUE... ?)
Yes, it would be technically possible - but it would require additional R/W storage and more resources than anyone would probably be willing to put into development.
This is what Raspberry PI will be for. It'll run a decent Linux distro, and you will be able to just plug in keyboards and mice, and an HDMI monitor, but still on a tiny bare-board $25 computer.
Can't wait to put my hands on one... 
@dxw00d
Wow! Wikipedia says January 2012 availability - I assume that is being slipped... Do you have any idea when they'll be available? It looks really interesting!
According to http://www.raspberrypi.org/ they are shipping now.
Ok - but are they taking orders? I created an account, but can't find an order-taking page. 
The shop offers only stickers for now...
(sigh) Bookmarked. :.
They've only just started manufacture. Hopefully it'll be available soon. It may be they are waiting for the whole first batch.
At least the videos look promising...