The reason it seems like something is "missing" is because you're thinking of it like it's finite. Infinity doesn't work the same way. There is no missing piece, because, as soon as you try to subtract that "missing piece" from 1, you've made it a finite number, and your subtraction won't result in 0.999.... Your subtraction will result in a number that is finite. Infinity is just really weird, and not at all intuitive. For example: most people think that there is nothing bigger than infinity, but there are actually some infinities that are bigger than others (they're called different levels of infinity), and this leads to things like the fact that there are the same "number" of positive integers as there are positive AND negative integers combined, but there are "more" irrational numbers than there are positive and negative integers. Trying to use intuition just doesn't work, because all of our intuition is based on finite things, since, as you said, most of our experience is finite.

However, that doesnt' mean that infinity doesn't exist in a physical world. There are plenty of ways in which infinity manifests itself in a physical world. Time is probably the most common; there is also the age old problem of continuously traveling half the distance to a destination (if you always travel half the remaining distance, it will take you an infinite amount of steps to get there).

Infinite series usually doesn't fly on videogame forums though

Maybe we should have a Pi counting competition thread on this website. Who can count the most digits in a row?

There is no part missing, that's the principe of an infinite series of numbers. 

Again and again, this is not a debate, it is a fact. The whole mathematics community agrees on that point, and there are lots of ways to prove it with any type of mathematics you want. It is not rounding it upwards. 

Read the Wiki page, really, everything is explained there, even the applications.

Or else we will come again to :

The lower primate in us still resists, saying: .999~ doesn't really represent a number, then, but a process. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart. Nonsense.^[44]

Well, it depends how you interpret things. We can't comprehend infinity because in my knowledge infinity doesn't exists.

So you're not a mathematician, and if you really are, you should ask some questions to your original college ;)

Just hope to god that there are some college/university math educated around these boards ...

You're working with infinite decimals so it's not the same. Feel free to write 0.99999.. on your math exam though!

From the same article:

Although the real numbers form an extremely useful number system, the decision to interpret the notation "0.999…" as naming a real number is ultimately a convention, and Timothy Gowers argues in Mathematics: A Very Short Introduction that the resulting identity 0.999… = 1 is a convention as well:

However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.[47]

A convention is the perfect word for it.  Plenty of examples are given where this is not accepted across all math systems.  That "lower primate" quote seems extremely narrow minded.