Sure, no problem running without an operating system, I do that kind of thing daily...
http://sam7stuff.blogspot.com/
You programs are, at least at first, not going to resemble desktop applications, I would avoid any libraries C libraries, no printfs or strcmps or things like that until you get the feel for it and find the right tools. No floating point as well. add some numbers do some shifting blink some leds.
codesourcery lite is probably the fastest way to get started, the gnueabi one I believe is the one you want.
This winarm site has a compiler and tons of non-os projects for seems like every arm based microcontroller.
http://www.siwawi.arubi.uni-kl.de/avr_projects/arm_projects/
Atmel is very very good about information, no doubt they have example programs you can try as well on the eval board.
emdebian is another cross compiler that is somewhat up to date and has binaries. building a gcc from scratch for cross compiling is not bad at all. The C library is another story though, and even the gcc library for that matter. I find it easier to do without either library.
It is possible get a C library working and run a great many kinds of programs. Depends on what you are looking to do. Ahh, just looked at the specs, that is a pretty serious eval board, plenty of power for an operating system should you choose to run one. You can certainly run programs that use the display as a user interface. read/write sd cards, usb, basically everything on the board, without an os, if you choose.