On the line of “what does the algebraic closure of $\mathbb{F}_p$ look like?”, it is also probably worth pointing out that, for $p=2$, this can be seen as the set of ordinals less than $\omega^{\omega^\omega}$ under nim addition (=exclusive or) and nim multiplication. But I'm afraid the order topology on $\omega^{\omega^\omega}$ will be of very little interest in answering James's question.
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