I want to upgrade Xcode but it tells me I first need to upgrade to Mac OS X 10.6.6 (I have 10.5.8).
However, the links I find online all seem to expect me to use Leopard or Lion.
How to I get there (without buying something)?
Unfortunately, you cannot. Due to the requirement of a 64-bit processor, a 64-bit OS, and the general evolution of requirements, much of Apple's software (and really, new software in general) simply reaches a point where you can't run newer versions without upgrading the OS, and occasionally your hardware, as a whole.