The notation for real numbers is $\mathbb{R}/\mathbf{R}$, integers: $\mathbb{Z}/\mathbf{Z}$, complex numbers $\mathbb{C}/\mathbf{C}$, and rational: $\mathbb{Q}/\mathbf{Q}$ but is there an agreed-upon way to express all numbers with similar notation?
Basically, if I were told to write a phrase that captured every number in existence, how would I do this?
Answer
As you observed, blackboard bold is a standard font used for successive extensions of number systems:
$$
\Bbb{N} \subseteq \Bbb{Z} \subseteq \Bbb{Q} \subseteq \Bbb{R} \subseteq \Bbb{C}
$$
The set of quaternions, denoted by $\Bbb{H}$ in honour of the mathematician W. R. Hamilton, would be the next step. The next extension is the set of octonions, denoted by $\Bbb{O}$ and the next one the set of sedenions, denoted by $\Bbb{S}$.
You will find many other extensions in the Wikipedia articles on
Hypercomplex numbers,
Hyperreal numbers and
Surreal numbers. The class -- this is no longer a set -- of all surreal numbers is denoted by the symbol $\mathbf{No}$. They are the largest possible ordered field: every other ordered field can be embedded in the surreals. Finally, a surcomplex number is a number of the form $a+bi$, where $a$ and $b$ are surreal numbers. Surcomplex numbers form an algebraically closed field (except for being a proper class). See also this question on Mathoverflow.
Finally, in answer to your final question "If I were told to write a phrase that captured every number in existence, how would I do this?", you could try the following aphorism:
If you ask me whether there is an ordered set of all numbers, the answer is no, but if you ask me whether there is an ordered class of all numbers, the answer is $\mathbf{No}$.
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