Notations such as $2$, $3$, $\mathord{\bullet} - \mathord{\bullet}$, $\dfrac{\mathord{\bullet}}{\mathord{\bullet}}$ (where $\mathord{\bullet}$ stands for an expression) have a universally-accepted meaning which has a precise mathematical definition: the number two, the number three, subtraction, division. I'm not going to state those precise definitions here, the important thing is that these definitions exist. (There are in fact many possible definitions, and mathematicians might disagree on which one is the “best” one, but the important point is that they all agree that whichever definition they've picked, the other mathematicians' definitions are equivalent.)

On the other hand, the notation $\ddots$ (called ellipsis) does not have a precise mathematical meaning. When you use this definition, it's up to you to make sure that there is only one possible interpretation.

The ellipsis is often a convenient shortcut, but you need to be careful when using it. If you write
$$
1^2 + 2^2 + 3^2 + \ldots + n^2
$$
then everybody will figure out that you mean the sum of the squares of the $n$ first integers, which could be written formally as $\sum_{i=1}^n i^2$. If you write
$$
\color{darkred}{1 + 4 + \ldots + n^2}
$$
then it's still a good guess, but it's a guess which is not straightforward, so you should avoid it. And if you write
$$
\color{red}{1 + \ldots + n^2}
$$
then there's more than one sensible interpretation ($\sum_{i=1}^n i^2$ or $\sum_{i=1}^{n^2} i$?), so this is not a good notation at all.

With a nested expression like yours, the notation means “the limit of the sequence with increasing levels of nesting”. This definition only works if, for all the ways to split the nesting, the expression converges to the same limit. Otherwise the definition is ambiguous, so the expression does not denote a number at all.

Let's use one particular way of parsing the expression as nested pieces: take the sequence
$$
2, \quad \cfrac{2}{3 - 2}, \quad \cfrac{2}{3 - \cfrac{2}{3 - 2}}, \quad \ldots
$$
which can be defined more formally as
$$
\begin{cases}
x_0 &= 2 \\
x_{i+1} &= \dfrac{2}{3-x_i} \\
\end{cases}
$$
This sequence is constant: all terms have the value $2$, and in particular it converges to $2$.

Now let's we cut differently:
$$
\cfrac{2}{3}, \quad \cfrac{2}{3 - \cfrac{2}{3}}, \quad \cfrac{2}{3 - \cfrac{2}{3 - \cfrac{2}{3}}}
$$
which is more formally defined by
$$
\begin{cases}
y_0 &= \dfrac{2}{3} \\
y_{i+1} &= \dfrac{2}{3-y_i} \\
\end{cases}
$$
Here we have a sequence that converges to $1$.

Since cutting the expression in two different ways results in different limits, the expression
$$ \cfrac{2}{3 - \cfrac{2}{3 - \cfrac{2}{3 - \cfrac{2}{\ddots}}}} $$
is ambiguous. It does not define a number.

How do we know which limit to reject? There are two possible answers.

- The notation is ambiguous, therefore the expression is invalid. We must reject both.
- The notation is ambiguous, therefore it's up to the author to clarify. You tell us which one to accept.