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The proof-theoretic ordinal of $\mathsf{EFA}$ and $\mathsf{RCA}_0^*$ are $\omega^3$ and the one of $\mathsf{PRA}$, $\mathsf{I\Sigma1}$, $\mathsf{RCA}_0$, etc. is $\omega^\omega$.

See https://ncatlab.org/nlab/show/ordinal+analysis.

In general, is there a connection between the fact that a system is considered finitist and its proof-theoretic ordinal? If so, is the bound $\leq\omega^\omega$ or $<\epsilon_0$? Would the bound for ultrafinitism be $<\omega$?

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    $\begingroup$ The meaning of "finitism" is unclear, so I do not expect there is a formalized way to answer to your question. $\endgroup$
    – Hanul Jeon
    Dec 3, 2023 at 18:13

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