The 0's don't give you anything. You could just say 1.

As for why they're equal, there are various proofs and explanations, but the simplest proof is probably:

1/3 + 1/3 + 1/3 = 1

0.333... + 0.333... + 0.333... = 1

0.999... = 1

asymptotically close as you add 9s, but 9 repeating means you DO add an infinity of 9s, so it equals 1.

The reasoning that persuaded me initially was 1/3 is .333 repeating, 2/3 is .666 repeating, and 3/3 is .999 repeating.

All of the answers given at this point seem to rely on intuition and how things "should" work. Intuition like that is great, except when trying to approach something that seems like a paradox when applying just intuition.

An expression like "0.999 repeating" does not just fall from the sky, after which we go ahead and probe it to figure out what it is or means or what it's equal to. A mathematical construction is precisely defined. So one goes to the definition and asks: what does "0.999 repeating" mean? After several rounds of unpacking definitions and verifying certain properties of convergent real sequences, one arrives at the conclusion that "0.999 repeating" is in fact 1. They are one and the same. Not "asymptotically", not "approximately" – they are the same. They are just written out differently. Your and mine handwriting is probably different, but that doesn't mean the number 1 written by one of us is different from the number 1 written by the other.

This is a typical misunderstanding about mathematics. Everything in mathematics is defined by humans, and the definitions can be unpacked layer by layer. The natural world is full of things that just exist, fully formed, without human interaction. The natural sciences give us wonderful tools to probe those things and figure out what they are and how they are. That's excellent, but it's usually not the right tool for answering questions like in this thread.

This is likely oversimplified, but an intuitive approach is:

1/3 = 0.333 repeating

3/3 = 0.999 repeating

1 = 0.999 repeating

If two numbers are different, you can always point to a different number between those two numbers.

So what number is between .999 repeating and 1?

Asymptotic analysis is only relevant if you have some series (or a function).

The series 0.9, 0.99, 0.999,... is asymptotically close to 1 and also asymptotically close to 0.9999... (with infinite 9s), since for any epsilon, I can find an index N after which all elements of the series are within epsilon of the target.

Since a single series can't have two limits, 1.0 should be equal to 0.999...

Note that real numbers are allowed to have infinite digits after the point, otherwise they wouldn't include things like 1/3.

My favorite version:

    x = 0.999...
    x - x/10 = 0.9
    x = 1

You have probably seen these already, but just in case: I think you would really enjoy p-adic numbers: https://en.wikipedia.org/wiki/P-adic_number