That's a damn good question. I think part of it depends on what your topic / area of interest is. That is, some fields are easier (cheaper / more accessible) than others. Doing novel work in high energy particle physics as an independent is probably damn tough, since you probably can't afford a particle accelerator. OTOH, if you're doing something where all you need is a computer (or many computers even), like, say, researching novel applications of genetic algorithms, then you can probably accomplish a lot without a ton of financial backing.
I hope this thread catches on and generates some activity, as I'd really like to hear what other HN'ers have to say on this topic. I'd say more about some stuff I've learned recently, but it's bed-time here.