Can we have nonstandard/transfinite "decimal" expansions? Can we take the decimal (or any arbitrary base) representation of a real number and just append some more digits beyond it? Is there a theory that covers this, maybe some kind of non-standard analysis?
E.g.:
$$x=0.x_1x_2x_3\ldots\in[0,1]$$
Let's just append some more digits:
$$x^*=0.\{x_1x_2x_3\ldots\}\{y_1y_2\ldots\}\{\ldots\}\{\ldots\}\ldots\in \text{ a non-standard version of }[0,1] $$
where $\{y_1y_2\ldots\}$ and each subsequent $\{\ldots\}$ is just another infinite string of digits.
So $y=0.\{\overline0\}\{y_1y_2\ldots\}$ would be like an infinitesimal, and we could say something like $0<y<\epsilon$ for every real number $\epsilon>0$. The $y_i$ digit string could be a non-standard infinitesimal base expansion.
Of course, I haven't claimed this is well-defined or defined at all. I'm not at all certain if such an idea can even be made sensible. I'm asking if there is some similar mathematical structure that is tractable and has been studied.
 A: Yes, you can - in fact, it's forced upon us in nonstandard analysis, and leads to a version of "$0.9999...=1$" remaining true despite the existence of infinitesimals.
Specifically, let's recall the definition of decimal expansions: something like $$0.a_1a_2a_3...$$ is really shorthand for $$\sum_{i\in\mathbb{N}}a_i\cdot 10^{-i},$$ which in turn is short for $$\lim_{n\rightarrow\infty}\sum_{i\in\mathbb{N}_{<n}}a_i\cdot 10^{-i}.$$ (Note the slightly odd summation notation - that's deliberate and will come up below.)
Now in nonstandard analysis we embed the usual reals $\mathbb{R}$ (= the unique-up-to-isomorphism connected ordered field) inside a larger structure ${}^*\mathbb{R}$ we call the hyperreals.

*

*Actually, there's no such thing as "the" hyperreals. We instead have a type of thing called a hyperreal field, of which there are many (even up to isomorphism). Almost always, the particular hyperreal field doesn't matter and so we just tacitly pick one, but this is technically an abuse of terminology.

But a particular case of this is that we also wind up embedding $\mathbb{N}$ inside its "nonstandard analogue" ${}^*\mathbb{N}$, and as  a consequence just as $\mathbb{N}$-indexed sums make sense in $\mathbb{R}$ so too do ${}^*\mathbb{N}$-indexed sums make sense in ${}^*\mathbb{R}$. These are exactly your "long" decimal expansions. And we get a lot of similarity: just as we have $$\sum_{i\in\mathbb{N}}9\cdot 10^{-i}=1\quad\mbox{in $\mathbb{R}$},$$ so too do we have $$\sum_{i\in{}^*\mathbb{N}}9\cdot 10^{-i}=1\quad\mbox{in ${}^*\mathbb{R}$}.$$

Stepping back a bit, the existence of ${}^*\mathbb{R}$-analogues of $\mathbb{R}$-flavored notions in general - in a technical sense - is one of the defining properties of a hyperreal field! Hyperreal fields are not just non-Archimedean ordered fields containing $\mathbb{R}$, they have to "form good analogies" in a precise sense.
Meanwhile, $\mathbb{N}$-indexed sums don't work at all in ${}^*\mathbb{R}$: they (at best) point to infinitesimal neighborhoods rather than specific elements. So there's actually a trade-off: we gain ${}^*\mathbb{N}$-indexed sums but lose $\mathbb{N}$-indexed sums. I've said a bit more about this here. (Perhaps surprisingly this turns out to be a consequence of the "good analogies" principle mentioned in the previous paragraph, but this gets a bit involved; if you're interested, the key term is "overspill.")
