Philosophers do spend a good deal of time worrying about this matter. Indeed, it is characteristic of many areas of philosophy to be particularly interested in the "unprovable assumptions" with which arguments begin. Two examples:

1. Perceptually-based beliefs---such as that there is a window in front of me---form the starting point for many of our beliefs. (Empiricists hold that all beliefs must be grounded there, but let's set that aside.) But it seems clear, at least to some of us, that these beliefs are not reached by argument from other beliefs. In that sense, they cannot be "proved" on the basis of anything else. How then should we understand how we arrive at such judgements? What is it for one of them to count as known? These are basic questions in the philosophy of perception.
2. In mathematics, theorems are proven from axioms. Axioms, on the other hand, are accepted as true without proof. On what ground do we accept such axioms as, say, that, if there are two sets A and B, then there is a set that is their union? (Perhaps one thinks this claim can be proven from other assumptions, but then of course we can just ask the same question about those assumptions.) Is the assumption simply arbitrary? Can we start with any axioms we like? That doesn't seem plausible. So do we have reasons for it? If so, what kinds of reasons could those be? These sorts of questions are common in the philosophy of mathematics.

Maybe that's not the kind of thing you had in mind. Another thing you might have meant is: Why bother giving an argument for something if the argument has to begin with assumptions for which you can't argue? Answer: An argument is supposed to show that, if you accept certain assumptions, then you must (or, perhaps, should) accept a certain conclusion, on pain of being irrational. The argument will be effective against anyone who accepts the assumptions. Whether the assumptions can be "proven" is neither here nor there. Of course, there's another question to be asked about why one should rational, but that's another matter.

The answer to the question about the meaning of life, of course, is "42".