This is the world of real numbers. What you describe with your apples is the realm of integers, and mathematically, these are different things.
What you are describing with the repeating fraction is an asymptotic limit, where you will get closer and closer to convergence, but will never actually reach it.
Adding more 9's to the repeating decimal only makes the number closer to one; as more are added, the closer the fraction is in value to 1. It really only ever approximately equals 1, but as an infinite number of 9's are added, in practical terms, it does equal 1, as the minute differences between an infinite number of 9's and an infinite number of 9's minus a single 9 is moot.