[E]very belief is based on assumptions, and those assumptions are based on other assumptions, which are based on others, and so on, until we are left with assumptions that we simply accept even though we have no justification for doing so.
This is known as the 'regress' argument, and there are several ways to resolve it. Firstly, one could follow Amanda in suggesting that the regress stops at unjustified beliefs. This would be epistemological nihilism - a denial of genuine justification.
A second option is to say that the regress stops at intrinsically justified (or 'self evident') beliefs. This is called foundationalism. The main problem with this view is that it isn't true. Nothing is self-evident when taken in isolation, but only when considered in relation to other assumptions that we hold. Perhaps we can be certain of how things seem to us (but perhaps not!), but even then, it's difficult to see how one could ground beliefs about the external world on such a subjective foundation.
The final option is to never stop the regress. Coherentists consider the structure of justifiction to be circular rather than linear. Quine speaks of a mutually-reinforcing 'web of beliefs', whereas Everitt & Fisher (Modern Epistemology, pp.102-3) prefer the analogy of a crossword puzzle:
In the image of the crossword, the answers in the crossword form the corpus of our beliefs, each individual answer that we fill in representing a single belief. When we get an answer that fits with an answer that we have already, that fact helps to confirm the correctness of the second answer [...] to some degree. But it is equally true that the second answer helps to confirm the correctness of the first answer. There is no asymmetry here in terms of confirmation [...] Notice, too, that the more answers which we get that fit in with the answers which we already have, the better confirmed we regard all of the answers, both the original set and the later additions.
So, on this view, we're never left with 'unsupported assumptions'. Every belief of mine is justified to the extent that it coheres with all the others.
When I pointed this out, Amanda made the interesting response:
Within a Coherentist framework, aren't you still accepting the assumption that the Coherentist framework is the right one? Sure, you might manage to get all of your other beliefs off the hook - but you still need to assume at least that one belief in Coherentist style justification.
Here we need to make the crucial distinction between what I believe about the nature of justification, and what is actually true about the nature of justification. If coherentism about justification is true, then it doesn't matter whether I assume it or not. The fact remains that my beliefs will be justified to the extent that they cohere with each other.
Now, we can ask whether my belief in coherentism is justified. I can straight away point out the fact that it coheres nicely with my other beliefs. So if coherentism is in fact true, then my belief here qualifies as a justified one. If coherentism is not in fact true, then I don't know whether my belief in it would be justified or not - I guess that would depend on what alternative account of justification happens to be true.
The important point to note here is that for my beliefs to be justified, I don't need to assume coherentism is true. It simply must be the case (in objective fact) that coherentism is true.