There are many ways to see that $0.999\ldots=1$ over the Reals (or over $\Bbb Q$ or $\Bbb C$ for that matter) like "Is it true that 0.99999…=1?", and the reasoning is easy enough:

If $x=0.\bar9$ then $10x=9.\bar{9}$ and consequenty, $9x=9.0$.

Some time ago I watched a math video which explained that the equality is not true everywhere, without mentioning what would break down, and no specific example was given, but what can go wrong, what are explicit examples?

What comes to mind are fields like the $p$-adics, the Superreals and Hyperreals. While I can follow their basic definitions or popular introductions, I have absolutely no intuition, let alone knowledge, what makes them different or not.

To make sense of $$s_n:=9\sum_{k=1}^n 10^{-k}$$ it's enough to have the arithmetic of a field. To make sense of $$0.\bar 9 := \lim_{n\to\infty} s_n$$ dunno what's needed here. A metric perhaps like $\operatorname{abs}:x\mapsto |x|$. The sequence should converge and the concept of convergence should make sense of course, and the limit should be $1$. Topologically closed is maybe too strong, but if the field is also a complete metric space that should suffice?

What I don't know is whether the above structures match (except for $\bar{\Bbb Q}_p$, which is however becond by comprehension / intuition).

And suppose the series $s_n$ from above converges to an element $0.\bar 9$ of the domain. Does this imply that $0.\bar 9 -1$ is an infinitesimal?

Maybe someone can iterate on these structures or on other domains that are worth talking about. And is it a requirement that $10^{-n}\to0$? Or may it be the case that $10^{-n}$ need not to become arbitrarily small and $0.\bar9$ still makes sense? What about Duals and split-complex?

**Note: My questions are not about uniqueness of representation.**