In biology, almost everything not genetics or biochemistry is easy to enter. You can easily (as in: focus and persevere) become the world expert in, say, spider or mushroom species in your local area and possibly discover some new species, or study the behavior of some animals in the wild and publish new findings. It's unlikely you will find a huge audience for it that way, but that's not to be expected, even if you work at a well-funded institution.
And it is possible to become somewhat well-known using this route. For a (semi; he has an education in biology) example, see https://en.m.wikipedia.org/wiki/Kees_Moeliker
You can also fairly easily go on expeditions such as ones trying to find out whether species X truly is extinct. It mostly requires not being asocial, being truly interested in the goals of the expedition, and paying for it yourself (see it as a holiday)
Other fields probably are similar: look where the money goes, choose a different subject area, and it is likely you can contribute.
Math is the ideal place for the independent outsider. The "group" mathematicians are just pushing the current fields to their logical conclusions, nothing wrong with that but its unlikely to create a new area of math or find anything profound. Its like mathematical engineering, using the tools/ideas that are known, and a safe career path. If an outsider has a proof people will listen.
The only other area I can think of would be computer science and inventing a true AI and not an AI mimic like Microsoft's recent twitter embarrassment, but in my view that would really be a development in epistemology with AI as its first application.
So do you think you are a genius? How old are you?
Attempts to quantize general relativity abound. A handful which anyone interested will mention off the top of their head: http://www.forbes.com/sites/startswithabang/2015/12/17/what-...
The problem is not just conceptual or mathematical, it's a total lack of experimental data. Even if you write down the correct theory of quantum gravity, you have no realistic way to test it - but you can rest assured that everybody pursuing a different approach will be right there telling you that you are an idiot.
There is also something to be said for analyzing all of the new datasets coming online and trying to gather new insights (this is also the least capital-intensive option!)
It just happens that most people who achieve a lot on their own usually stay out of the limelight or public eye. This is especially true in ecology or biology.