I'm attempting to evaluate the second element of the fundamental sequence for the infinite ordinal known as $\zeta_0$.
$\zeta_0$ is defined as the supremum of: $\{\varepsilon_0,\varepsilon_{\varepsilon_0},\varepsilon_{\varepsilon_{\varepsilon_0}},...\}$, where $\varepsilon_{n+1}$ is defined as the supremum of: $\{\varepsilon_n,\varepsilon_n^{\varepsilon_n},\varepsilon_n^{\varepsilon_n^{\varepsilon_n}},...\}$
$\varepsilon_0$, in turn is defined as the supremum of: $\{\omega,\omega^\omega,\omega^{\omega^\omega},...\}$
And omega is the infinite ordinal which is the supremum of the natural numbers: $\{1,2,3,...\}$
I've given the evaluation of the second element of this ordinal's fundamental sequence (For the purposes of a theoretical evaluation of the fast growing diagonalization hierarchy.)
The attempt here is what I have so far:
$\zeta_0=\sup\{\varepsilon_0,\varepsilon_{\varepsilon_0},...\}$
$\varepsilon_{\varepsilon_0}=\sup\{\varepsilon_\omega,\varepsilon_{\omega^\omega},...\}$
Skipping a few easy ones, we arrive at:
$\varepsilon_{\omega+2}=\sup\{\varepsilon_{\omega+1},\varepsilon_{\omega+1}^{\varepsilon_{\omega+1}},...\}$
$\varepsilon_{\omega+1}^{\varepsilon_{\omega+1}}=\sup\{\varepsilon_{\omega+1}^{\varepsilon_\omega},\varepsilon_{\omega+1}^{\varepsilon_\omega^{\varepsilon_\omega}},...\}$
$\varepsilon_{\omega+1}^{\varepsilon_\omega^{\varepsilon_\omega}}=\sup\{\varepsilon_{\omega+1}^{\varepsilon_\omega^{\varepsilon_0}},\varepsilon_{\omega+1}^{\varepsilon_\omega^{\varepsilon_1}},...\}$
$\varepsilon_{\omega+1}^{\varepsilon_\omega^{\varepsilon_1}}$ gives $\varepsilon_{\omega+1}^{\varepsilon_\omega^{\varepsilon_0^{\varepsilon_0}}}$, which gives $\varepsilon_{\omega+1}^{\varepsilon_\omega^{\varepsilon_0^{\omega^\omega}}}$, which gives $\varepsilon_{\omega+1}^{\varepsilon_\omega^{\varepsilon_0^{\omega+2}}}$.
This is where I'm stuck.
I have made several attempts to move on from this point. One of which lead to this mess:
$\varepsilon_{\omega+1}^{\varepsilon_\omega^{\omega^4+8*\omega^3+24*\omega^2+32*\omega+16}}$
But I feel like I made several bad assumptions and missteps to arrive at that. At one point along the way, I took $\varepsilon_0^{\omega+2}$ and turned it into $\varepsilon_0^\omega*\varepsilon_0^2$, then evaluated the first factor to $\varepsilon_0^2$, and turned the whole thing into $\varepsilon_0^4$.
This felt wrong and I'm pretty sure it is.
A fundamental sequence for a limit ordinal is simply an increasing sequence of ordinals whose supremum is the aforementioned ordinal. Unfortunately, there is not a single definition of fundamental sequence for all ordinals, but for $\zeta_0$ it would be natural to set
$\zeta_0 [0] = 0$
$\zeta_0 [n+1] = \varepsilon_{\zeta_0 [n]}$
So $\zeta_0 [2]$ would be $\varepsilon_{\varepsilon_0}$. However, it seems like what you want is to keep taking the second element of the fundamental sequence until you get to a successor ordinal. This can be done, but you need to make sure that you have predefined your fundamental sequences in a consistent way. For each ordinal less than or equal to $\zeta_0$, you want a single fundamental sequence, so you need to be sure that two different expressions for the same ordinal don't generate two different fundamental sequences. Towards that end, it is helpful to have a canonical form for every ordinal up to $\zeta_0$.
Here is one way to define a canonical form: For each ordinal $\alpha$, write it as a sum of additively principal ordinals (that is, powers of $\omega$): $\alpha_1 + \alpha_2 + \ldots + \alpha_n$. If $\alpha$ is additively principal but not an $\varepsilon$-number, write it as $\omega^\beta$. If $\alpha$ is an $\varepsilon$-number but not a $\zeta$-number, write it as $\varepsilon_\beta$. If $\alpha$ is a $\zeta$-number, write it as $\zeta_\beta$. That will give you a single expression for each ordinal, so you can define fundamental sequences consistently.
Your fundamental sequence values seem a little inconsistent; It seems that you are using the rule $\varepsilon_\alpha [n] = \varepsilon_{\alpha[n]}$, which is a good rule, but then you have $\omega[2] = 2$ but $\varepsilon_\omega [2] = \varepsilon_1$, which seems inconsistent. I would set $\varepsilon_\omega [2] = \varepsilon_2$.
Other than that, your evaluations seem fine, but you were right that you made a mistake but setting $\varepsilon_0^{\omega + 2} [n] = \varepsilon_0^{n+2}$. The problem is that $\varepsilon_0^{n+2}$ has limit $\varepsilon_0^{\omega}$, not $\varepsilon_0^{\omega + 2}$. In general, the function $f(\alpha) = \alpha \beta$ is not continuous, so setting $(\alpha \beta) [n] = (\alpha [n]) \beta$ does not work in general. The function $f(\beta) = \alpha \beta$ is continuous, so setting $(\alpha \beta) [n] = \alpha (\beta[n])$ will work. However, remember to express each ordinal in a canonical form and define a specific set of rules for your fundamental sequences.