That's actually an excellent question, meldraft!
Unfortunately, I'm merely a mathematician who knows nothing about physics, but I do think I can give you a satisfactory answer.
Firstly, completeness of a normed space isn't really essential. If we want to, we can formulate all of the mathematical and physicsal theories in incomplete spaces. OK, the formulation will be a lot more complicated, but we can still do it. Why is it true what I said? Well, every normed space (metric space even) has a unique completion. That is, it is possible to adjoin a few elements to an incomplete space in order to make it complete. Does, it is possible to work in incomplete spaces and occasionally refer to it's completeness whenever we need it.
That said, the only reason we work with \mathbb{R} is because the space is complete. Let me elaborate. Say we want to do a measurement of some kind. For example, say we want to measure the lengths of persons. Then possible outcomes will be 1.90m, 1.85m, 1.77m, etc. As you see, all the outcomes will be rational numbers. So for measurement purposes, we can work in \mathbb{Q} most of the time.
Why do we consider those measurements in \mathbb{R}, then? Well, for the simple reason that we want to do calculus on the measurement. For example, we want to find a curve that best fits our measurements and we want to find the integral of that curve. But the integral is defined as a limit! But, when working in \mathbb{Q}, we have no reason why this limit should even exist! So, in order to make things work, we need to be able to talk about limits. Not that we're really interested in the value of the limit (we will approximate that value anyway), but to make things easy, we want to know that the limit exists. And this is where completness comes in.
Approximating zeroes of a function is another thing we need completeness for. Say we have an equation x=cos(x)+sin(x). One method to approximate the root is to form the function
F:\mathbb{R}\rightarrow\mathbb{R}:x\rightarrow \cos(x)+\sin(x)
Applying this function to an arbitrary value will eventually converge to the root. So F(F(F(F(0)))) will be a close approximation to the root. Of course, the approximation is enough for all practical purposes, but to make the theory easier, we will need that the sequence exist.
Another thingy involves differential equations. In practise, we will always solve differential equations by the computer and we will find approximations for them. But to make our theory simple, we need to know that the differential equations have solutions. But this is exactly where completeness comes in! Given an differential equation, we can always transform it into an integral equation, for example:
f(x)=g(x)+\int_0^x{f(t)dt}
How do we show that we have a solution, well, we form the function
F:\mathcal{C}[a,b]\rightarrow \mathcal{C}[a,b]:f\rightarrow g+\int f
Iterating this function (that is, doing F(F(F(F(F(F(0))))))) will eventually converge to the desired function. But to know that the limit even exists, we need to know that the space C[a,b] is complete!
To summarize, I feel that completeness is actually a technical requirement to make the theory easier. We can do without it, but the theory will become much uglier!
