Does there exist a real number between 0.999.... and 1?  If there is, find it.  If there isn't, then 0.999... = 1.  This is because between any two different real numbers, there must exist another real number between them.

Another good proof is 1/3 = 0.333.... * 3 = 0.999.... but 1/3 * 3 = 1.

I've spent a good 5 hours on this intriguing puzzle, reading the different links and past discussions and the such.  I personally am not as adept at mathematics as some of you clearly are, but I would say that from all this my conclusion is that 0.999... isn't = 1 and in the original question the problem lies with when you multiply 0.999... by 10 to get 9.999... The reason being is that you just can't do it to start off with.  When you multiply something, you first multiply the very most right value of the number, right?. For example 3 multiplied by 1.45678, you would first multiply 3 by 8 carry the 2 etc.  In this case there is no very most right value as it is infinite and hence you cant just multiply it by 10 to get 9.999...
We always take the shortcut to move the decimal place, but is that a proven method or just a shortcut.  I suppose the fact that we are using a decimal system for our calculations would indicate that you may be able to, but I thought I'd just pose that as a question.
Now, I don't know if I'm just looking at this too simply or not, but that's just a thought.

Hi Ausbomba. Yes, it is a great discussion point!

My view on it is that infinity has no end, so there is no "end" to 0.999..., nor is there an end to 10×0.999...=9.999...

It is the peculiar nature of infinity. You see inifnity is not a big number! It is a concept that says "not finite", or simply "no end".