This question is based on the assumption that $V \ne L$ and we have $\omega_1^L < \omega_1$ (here $\omega_1^L$ is equal to the supremum of ordinals accidentally writable by no-oracle Ordinal Turing Machines).  
  
Consider Ordinal Turing Machines (called “$\omega_{\alpha}$-machines”) with an oracle that provides access to all [transfinite initial ordinals](https://mathworld.wolfram.com/InitialOrdinal.html).  
  
Any $\omega_{\alpha}$-machine is an Ordinal Turing Machine equipped with an extra tape (the _oracle tape_). We may assume that this tape is read-only.  
  
Let $t(\alpha)$ denote the symbol written on an $\alpha$-th cell of the oracle tape. Then $t(\alpha) = 1$ if and only if $\alpha$ is an initial ordinal.  
  
If $\epsilon > 0$, then the $\epsilon$-stabilization time of a machine is the least ordinal $\gamma_0$ such that the values of all symbols written on all cells of the initial segment of length $\epsilon$ of the output tape never change at any time $\gamma > \gamma_0$. If $\epsilon = 0$, then the $\epsilon$-stabilization time of a machine is the least ordinal $\gamma_0$ such that the values of all symbols written on _all cells_ (i.e. cells indexed by any element of the class of all ordinals) of the entire output tape never change at any time $\gamma > \gamma_0$. If a machine halts, then $\gamma_0$ is not greater than the halting time.  
  
Let $F_{\epsilon}(i)$ denote the $\epsilon$-stabilization time of an $i$-th $\omega_{\alpha}$-machine, assuming that all computations start with no ordinal parameters (i.e. empty input). Here we assume that if a corresponding machine diverges (i.e. the values of all symbols written on all cells of the initial output segment of length $\epsilon$, or all cells of the entire output tape if $\epsilon = 0$, do not stabilize), then $F_{\epsilon}(i) = 0$.  
  
Assuming that we have fixed a particular way to encode a countable ordinal by an infinite binary sequence of length $\omega_0=\omega$, the ordinal $\tau_0$ is defined as the supremum of ordinals _eventually writable_ by $\omega_{\alpha}$-machines with empty input on the initial segment of length $\omega_0=\omega$ of the output tape. The reasoning behind this definition of $\tau_0$ is that there may be $\omega_{\alpha}$-machines whose initial output segment of length $\omega$ stabilizes at a time $\ge \omega_1$ (i.e. $F_{\omega}(i) \ge \omega_1$), yet all other output segments are irrelevant: they stabilize at an arbitrarily large time or _even diverge_. That is, I suppose that there may exist a machine $M_n$ such that, for example, $F_0(n) = 0$, yet $F_{\omega}(n) \ge \omega_1$. In this case, if the eventually stable content on the initial $\omega$-segment of the output tape encodes an ordinal, this countable ordinal is eventually writable by $M_n$.
  
The ordinal $\tau_1$ is defined as follows: $$\tau_1 = \sup \{F_0(i) : i \in \mathbb{N}\}.$$  
  
Is it possible to estimate how large are $\tau_0$ and $\tau_1$ (at least, give a “reasonably accurate” estimate for the lower/upper bounds)? In particular, is $\tau_0$ larger than the least ordinal $\delta$ such that $L_{\delta} \prec_{\Sigma_3} L_{\omega_1}$ (the latter is mentioned in [this comment](https://mathoverflow.net/questions/259100/memorable-ordinals#comment639579_259105) and [this answer](https://mathoverflow.net/a/259500) on Mathoverflow)?