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An ordinal is the order type of a well-ordered set. The first few ordinals are $0, 1, 2, \dots, \omega, \omega+1, \dots$ where $\omega$ is the order type of $\mathbb{N}$, and $\omega+1$ is the order type of $\mathbb{N}$ together with a maximum element.
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How large is the supremum of halting times of Infinite Time Turing Machines, assuming that h...
This and related questions about decision times of ITTMs are addressed in this paper:
"Decision times of infinite computations"
https://arxiv.org/abs/2011.04942
The ordinal you are asking for seems to …