The problem with these equations is the "$\dots$". In infinite expressions such as the infamous $0.9999\dots$, the ellipsis represents a limit. In the decimal case, what it means is the limit of the sequence $$
0.9, 0.99, 0.999, \text{ and so on}
$$
which you can prove converges in various ways, for example it is trivially increasing, and is bounded above by $1$. The manipulations that follow are valid assuming the limit exists. Let $x=0.9999\dots$ (i.e., the limit of the above sequence). Then:\begin{eqnarray}
x &=& 0.9999\dots\\
10x&=&9.9999\dots\\
10x&=&9+x\\
x &=& 1
\end{eqnarray}
Now, we turn to your expression $999\dots$. The ellipsis here would indicate that this is meant to be the limit of the sequence $$
9, 99, 999, \text{ and so on}
$$
Of course, the limit doesn't exist under the standard metric on $\mathbb{R}$, but if you pretend you don't know that, you can do the kind of manipulations you wrote out and say that if the limit $999\dots$ exists, it is $-1$.
It looks very similar to an expression for a $p$-adic number, which are typically represented by an infinite string of base-$p$ digits for some prime $p$ (see this question and its answers for why it should be a prime). In $p$-adic spaces, you can do those sorts of manipulations you wrote down and make meaningful sense of them. Wikipedia presents the nice example of the $5$-adic $$
\dots1313132 = \frac13
$$
Indeed, you can represent any rational number as an infinite string of base $p$ digits. So, for example, in $5$-adics again, you can write $$
\dots 44444 = -1
$$
which follows from the same kind of manipulations you did: Let $x = \dots 4444$. Then (keep in mind I'm in base $5$ here):\begin{eqnarray}
x &=& \dots 4444\\
10x &=& \dots 4440\\
10x+4&=&\dots 4444 = x\\
4x +4&=&0\\
x &=&-1
\end{eqnarray}
