It is well-known that $\mathbf{PRA}$ plus $\epsilon_0$-induction cannot prove all $\mathbf{PA}$ theorems (essentially because the former is finitely axiomatizable while the latter isn't). Are there any concrete examples known (preferable "natural")?

(This is an exact copy of my question on MSE, but I expect I won't get any answer there, so I hope I'm allowed to cross-post it here)