The logic of Succession and Recursion is extending first order logic with equality by the following:
Add primitives of $N,0,S$ standing for "Natural, Zero, Successor". Also we add primitive function symbols $\mathcal R^{f,\alpha}$ for every function symbol $f$, and constant symbol $\alpha$.
Axiomatize:
Succession axioms: $N(0) \\ \forall x: N(x) \to N(S(x)) \\ \forall x: N(x) \to S(x) \neq 0 \\ \forall x \forall y : N(x) \land N(y) \land S(x)=S(y) \to x=y$
Recursion: if $f$ is a function symbol (primitive or parameter free definable), and $\phi$ a one place predicate symbol (primitive or definable), and $\alpha$ a constant symbol (primitive or definable), then:
$$\phi(\alpha) \land \\ \forall x: \phi(x) \to \exists y: f(x)=y \land \phi(y) \\ \implies \\ \mathcal R^{f,\alpha} (0) =\alpha \land \\ \forall n (N(n) \implies \mathcal R^{f,\alpha}(S(n)) = f(\mathcal R^{f,\alpha}(n)))$$
What's the exact consistency strength of this system?