This doesn't answer the question you asked because you seem to care about extensions of the coefficient field $\mathbb{F}_p$, but I think it's at least worth pointing out that, if $k := \mathbb{F}_p^{\mathrm{alg}}$ is the algebraic closure of $\mathbb{F}_p$, then the algebraic closure of $k(t)$ can be seen inside the field of Hahn series $k[\![t^{\mathbb{Q}}]\!]$ over $k$ which has at least some similarity to a topological completion. So it may interest you to look into that if you don't already know about such fields.
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