If I remove the power set axiom from ZF, but retain the axiom of infinity, what will be the largest ordinal that can still be proven to exist. Or, if no largest such ordinal exist, what is the limit ordinal of all ordinals that can still be proven to exist. Of course, it is possible that even in this extended sense no such largest ordinal exists.
As Asaf convinced me, there is no need to modify the axiom of replacement or the axiom of foundation for the question to make sense in this context. However, the question as initially intended was about a useful set theory without the power set axiom (i.e. a theory one would use occasionally). As I gather from Miha Habič's comment, this would have been Kripke-Platek set theory (plus the axiom of infinity), and the largest provably existing ordinal would be $\omega_1^{CK}$ in that case.