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One way to think about SVO is in terms of a function $F:(\omega_1)^\omega \rightarrow \omega_1$. For example, writing $\omega_1=w$, we will have $\mathrm{SVO}=\mathrm{sup}\{\,F(w^i) \,\, | \,\, 1 \leq i<\omega\}$. This is analogous to thinking $\Gamma_0$ in terms of $F:(\omega_1)^2 \rightarrow \omega_1$. So, we will have $\Gamma_0$ as the first fixed point of the ordinal function $x \mapsto F(\omega_1+\omega_1 \cdot x)$. Quite informally, I use the term "storage-functions" for these functions $F$. The $\omega_1$ isn't quite relevant in the sense that we just need an ordinal "big enough" ($\omega_{CK}$ would be sufficient in the above two cases). But anyway, that's besides the point. The point here is that when a function $x \rightarrow \omega^x$ alongside with a command of form $u:=\omega_1$ is given to us, then there is a specific infinite program which can compute the storage function (in input-output sense).

Is this relevant to your question? Yes. The same program that gives us SVO when given the function $x \mapsto \omega^x$ will take us to the "analogue of SVO" in the question. A similar observation applies to LVO (using the function $x \mapsto \omega_x$). But the issue of "storage function" seems to become trickier in this "analogue case".

EDIT: I am not suggesting to gloss over several important aspects (suchsuch as equivalence of thisdifferent and proabably many more depending on exactly what is being discussed) that should be considered more formally if we are being fully precise. I am only suggesting an equivalence (between $\Gamma_0$ and "analogue $\Gamma_0$" and other ordinals such as in question) that is very likely to holddefinitions. If we are being fully precise/detaileddetailed, I will admit the paragraphs above are quite insufficient. END

Finally, very briefly, towards the end you mention "extension" of transfinite variable. In the case of original hierarchy these kind of basic extensions would be handled by extending the domain of the "storage function" by a very modest amount. For example, from $(\omega_1)^{\omega_1} \rightarrow \omega_1$$F:(\omega_1)^{\omega_1} \rightarrow \omega_1$ to $(\omega_1)^{\omega_1} \cdot \omega \rightarrow \omega_1$$F:(\omega_1)^{\omega_1} \cdot \omega \rightarrow \omega_1$ etc. Similarly observations made earlier in this post about the "same" program taking us to the "analogue" of corresponding ordinal would apply (when given $x \mapsto \omega_x$ instead of $x \mapsto \omega^x$).

EDIT2: To OP (as a precaution): Please note that just writing $F:(\omega_1)^{\omega_1} \rightarrow \omega_1$ (or anything of that sort) doesn't mean that the underlying function has been fully well-defined and neither I meant to imply that. In the given specific cases, precise definition can either be descriptive or based upon a (infinite) program which computes the function (given an extra command of form $u:=\omega_1$). Showing that the given def. satisfy certain desirable/required properties is bound to be more work. END

One way to think about SVO is in terms of a function $F:(\omega_1)^\omega \rightarrow \omega_1$. For example, writing $\omega_1=w$, we will have $\mathrm{SVO}=\mathrm{sup}\{\,F(w^i) \,\, | \,\, 1 \leq i<\omega\}$. This is analogous to thinking $\Gamma_0$ in terms of $F:(\omega_1)^2 \rightarrow \omega_1$. So, we will have $\Gamma_0$ as the first fixed point of the function $x \mapsto F(\omega_1+\omega_1 \cdot x)$. Quite informally, I use the term "storage-functions" for these functions $F$. The $\omega_1$ isn't quite relevant in the sense that we just need an ordinal "big enough" ($\omega_{CK}$ would be sufficient in the above two cases). But anyway, that's besides the point. The point here is that when a function $x \rightarrow \omega^x$ alongside with a command of form $u:=\omega_1$ is given to us, then there is a specific infinite program which can compute the storage function (in input-output sense).

Is this relevant to your question? Yes. The same program that gives us SVO when given the function $x \mapsto \omega^x$ will take us to the "analogue of SVO" in the question. A similar observation applies to LVO. But the issue of "storage function" seems to become trickier in this "analogue case".

EDIT: I am not suggesting to gloss over several important aspects (such as this and proabably many more depending on exactly what is being discussed) that should be considered more formally if we are being fully precise. I am only suggesting an equivalence (between $\Gamma_0$ and "analogue $\Gamma_0$" and other ordinals such as in question) that is very likely to hold. If we are being fully precise/detailed, I will admit the paragraphs above are quite insufficient. END

Finally, very briefly, towards the end you mention "extension" of transfinite variable. In the case of original hierarchy these kind of basic extensions would be handled by extending the domain of the "storage function" by a very modest amount. For example, from $(\omega_1)^{\omega_1} \rightarrow \omega_1$ to $(\omega_1)^{\omega_1} \cdot \omega \rightarrow \omega_1$ etc. Similarly observations made earlier in this post about the "same" program taking us to the "analogue" of corresponding ordinal would apply.

One way to think about SVO is in terms of a function $F:(\omega_1)^\omega \rightarrow \omega_1$. For example, writing $\omega_1=w$, we will have $\mathrm{SVO}=\mathrm{sup}\{\,F(w^i) \,\, | \,\, 1 \leq i<\omega\}$. This is analogous to thinking $\Gamma_0$ in terms of $F:(\omega_1)^2 \rightarrow \omega_1$. So, we will have $\Gamma_0$ as the first fixed point of the ordinal function $x \mapsto F(\omega_1+\omega_1 \cdot x)$. Quite informally, I use the term "storage-functions" for these functions $F$. The $\omega_1$ isn't quite relevant in the sense that we just need an ordinal "big enough" ($\omega_{CK}$ would be sufficient in the above two cases). But anyway, that's besides the point. The point here is that when a function $x \rightarrow \omega^x$ alongside with a command of form $u:=\omega_1$ is given to us, then there is a specific infinite program which can compute the storage function (in input-output sense).

Is this relevant to your question? Yes. The same program that gives us SVO when given the function $x \mapsto \omega^x$ will take us to the "analogue of SVO" in the question (using the function $x \mapsto \omega_x$). But the issue of "storage function" seems to become trickier in this "analogue case".

EDIT: I am not suggesting to gloss over several important aspects such as equivalence of different definitions. If we are being fully detailed, I will admit the paragraphs above are quite insufficient. END

Finally, very briefly, towards the end you mention "extension" of transfinite variable. In the case of original hierarchy these kind of basic extensions would be handled by extending the domain of the "storage function" by a very modest amount. For example, from $F:(\omega_1)^{\omega_1} \rightarrow \omega_1$ to $F:(\omega_1)^{\omega_1} \cdot \omega \rightarrow \omega_1$ etc. Similarly observations made earlier in this post about the "same" program taking us to the "analogue" of corresponding ordinal would apply (when given $x \mapsto \omega_x$ instead of $x \mapsto \omega^x$).

EDIT2: To OP (as a precaution): Please note that just writing $F:(\omega_1)^{\omega_1} \rightarrow \omega_1$ (or anything of that sort) doesn't mean that the underlying function has been fully well-defined and neither I meant to imply that. In the given specific cases, precise definition can either be descriptive or based upon a (infinite) program which computes the function (given an extra command of form $u:=\omega_1$). Showing that the given def. satisfy certain desirable/required properties is bound to be more work. END

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One way to think about SVO is in terms of a function $(\omega_1)^\omega \rightarrow \omega_1$$F:(\omega_1)^\omega \rightarrow \omega_1$. For example, writing $\omega_1=w$, we will have $\mathrm{SVO}=\mathrm{sup}\{\,F(w^i) \,\, | \,\, 1 \leq i<\omega\}$. This is analogous to thinking $\Gamma_0$ in terms of $(\omega_1)^2 \rightarrow \omega_1$$F:(\omega_1)^2 \rightarrow \omega_1$. QuiteSo, we will have $\Gamma_0$ as the first fixed point of the function $x \mapsto F(\omega_1+\omega_1 \cdot x)$. Quite informally, I use the term "storage-functions" for these functions $F$. The $\omega_1$ isn't quite relevant in the sense that we just need an ordinal "big enough" ($\omega_{CK}$ would be sufficient in the above two cases). But anyway, that's besides the point. The point here is that when a function $x \rightarrow \omega^x$ alongside with a command of form $u:=\omega_1$ is given to us, then there is a specific infinite program which can compute the storage function (in input-output sense).

Is this relevant to your question? Yes. The same program that gives us SVO when given the function $x \mapsto \omega^x$ will take us to the "analogue of SVO" in the question. A similar observation applies to LVO. But the issue of "storage function" seems to become trickier in this "analogue" case"analogue case".

EDIT: I am not suggesting to gloss over several important aspects (such as this and proabably many more depending on exactly what is being discussed) that should be considered more formally if we are being fully precise. I am only suggesting an equivalence (between $\Gamma_0$ and "analogue $\Gamma_0$" and other ordinals such as in question) that is very likely to hold. If we are being fully precise/detailed, I will admit the paragraphs above are quite insufficient. END

One way to think about SVO is in terms of a function $(\omega_1)^\omega \rightarrow \omega_1$. This is analogous to thinking $\Gamma_0$ in terms of $(\omega_1)^2 \rightarrow \omega_1$. Quite informally, I use the term "storage-functions" for these functions. The $\omega_1$ isn't quite relevant in the sense that we just need an ordinal "big enough" ($\omega_{CK}$ would be sufficient in the above two cases). But anyway, that's besides the point. The point here is that when a function $x \rightarrow \omega^x$ alongside with a command of form $u:=\omega_1$ is given to us, then there is a specific infinite program which can compute the storage function (in input-output sense).

Is this relevant to your question? Yes. The same program that gives us SVO when given the function $x \mapsto \omega^x$ will take us to the "analogue of SVO" in the question. A similar observation applies to LVO. But the issue of "storage function" seems to become trickier in this "analogue" case.

One way to think about SVO is in terms of a function $F:(\omega_1)^\omega \rightarrow \omega_1$. For example, writing $\omega_1=w$, we will have $\mathrm{SVO}=\mathrm{sup}\{\,F(w^i) \,\, | \,\, 1 \leq i<\omega\}$. This is analogous to thinking $\Gamma_0$ in terms of $F:(\omega_1)^2 \rightarrow \omega_1$. So, we will have $\Gamma_0$ as the first fixed point of the function $x \mapsto F(\omega_1+\omega_1 \cdot x)$. Quite informally, I use the term "storage-functions" for these functions $F$. The $\omega_1$ isn't quite relevant in the sense that we just need an ordinal "big enough" ($\omega_{CK}$ would be sufficient in the above two cases). But anyway, that's besides the point. The point here is that when a function $x \rightarrow \omega^x$ alongside with a command of form $u:=\omega_1$ is given to us, then there is a specific infinite program which can compute the storage function (in input-output sense).

Is this relevant to your question? Yes. The same program that gives us SVO when given the function $x \mapsto \omega^x$ will take us to the "analogue of SVO" in the question. A similar observation applies to LVO. But the issue of "storage function" seems to become trickier in this "analogue case".

EDIT: I am not suggesting to gloss over several important aspects (such as this and proabably many more depending on exactly what is being discussed) that should be considered more formally if we are being fully precise. I am only suggesting an equivalence (between $\Gamma_0$ and "analogue $\Gamma_0$" and other ordinals such as in question) that is very likely to hold. If we are being fully precise/detailed, I will admit the paragraphs above are quite insufficient. END

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I am hoping an expert would answer this question so as to shed light on deeper or more profound points. As such, this is a basic answer covering some easy to understand points. This is based upon number of things I thought about years ago (it seems that some of those observations can be used in this question).

So let's start with your question "how far does this notation reach". I don't know what would be the answer to the question. It seems that to be able to answer though one would have to frame the question much more precisely (and I am not certain what that framing would be). Meanwhile the specific constructions you are posting (and far beyond that) are easily understood thinking in terms of generalized notion of being able to do complex calculations on ordinals.

For example, let's talk about something specific. In the beginning of your post you mention a way of starting with the function $x \mapsto \omega_x$ and how to arrive at an ordinal that is analogous to $\Gamma_0$. This analogy can be made precise using infinite programs that are sufficiently powerful. How so? Assume that a function $f:\mathrm{Ord} \rightarrow \mathrm{Ord}$ is "given" to the program. Exactly the same program that takes one to $\Gamma_0$ (given $f(x)=\omega^x$) will take one to "analogue of $\Gamma_0$" that you mention in your question. The only difference is that the function $f$ "given" to the program now is $f(x)=\omega_x$.

Now the same observations apply to bigger ordinals. I haven't studied the original Veblen paper so I am not 100% sure if the correspondences that I mention below are exact or not (so please correct if they aren't).

One way to think about SVO is in terms of a function $(\omega_1)^\omega \rightarrow \omega_1$. This is analogous to thinking $\Gamma_0$ in terms of $(\omega_1)^2 \rightarrow \omega_1$. Quite informally, I use the term "storage-functions" for these functions. The $\omega_1$ isn't quite relevant in the sense that we just need an ordinal "big enough" ($\omega_{CK}$ would be sufficient in the above two cases). But anyway, that's besides the point. The point here is that when a function $x \rightarrow \omega^x$ alongside with a command of form $u:=\omega_1$ is given to us, then there is a specific infinite program which can compute the storage function (in input-output sense).

Is this relevant to your question? Yes. The same program that gives us SVO when given the function $x \mapsto \omega^x$ will take us to the "analogue of SVO" in the question. A similar observation applies to LVO. But the issue of "storage function" seems to become trickier in this "analogue" case.

Finally, very briefly, towards the end you mention "extension" of transfinite variable. In the case of original hierarchy these kind of basic extensions would be handled by extending the domain of the "storage function" by a very modest amount. For example, from $(\omega_1)^{\omega_1} \rightarrow \omega_1$ to $(\omega_1)^{\omega_1} \cdot \omega \rightarrow \omega_1$ etc. Similarly observations made earlier in this post about the "same" program taking us to the "analogue" of corresponding ordinal would apply.

How time consuming it would be to write the detail of storage functions? For $(\omega_1)^2 \rightarrow \omega_1$ (starting with $x \mapsto \omega^x$) taking us to $\Gamma_0$ it should be fairly simple (though still a bit long to post all of it here). And then it gets lengthier, as it gets more complicated.