The answers already posted are quite satisfying, I'd just like to add one more point of view (at the risk of making the thing more confused for the OP :). When Sobolev started solving PDEs, he did not have reasonable function spaces available: working in $C^2$ is a nightmare as soon as you want to do calculus of variations, and it is immediately clear that 'something is missing'. You naturally construct solutions by approximating them (with minimizing sequences, with smooth approximations etc. etc.). The original approach of Sobolev was: well, all I have is this approximating sequence, so THIS SEQUENCE is my solution, whatever that means. This was his original definition of 'weak solution'.
As you see, he was dispensing completely with completeness, and working only with functions in a dense subspace. This is perfectly fine, and I'm tempted to answer to the original question with the paradox: completeness is not really necessary, even from a theoretical standpoint, since of course you can embed every normed space in a complete one. But this is very awkward; it is vastly more economical to 'define' the limit of your approximating sequence. Indeed, this procedure is precisely what is called completion. Working in a complete space makes it possible to take the limit of your approximation and define a solution as a concrete object. 100 yeasr later, we find this approach totally natural. I think this was one of the driving forces behind the universal adoption of complete spaces in analysis.
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