Now asked at MO.
Motivated by this question, I'd like to ask whether in a precise sense there are no "interesting" functions which are provably recursive in Robinson's arithmetic $\mathsf{Q}$.
For an appropriate theory $T$, say that an $n$-ary $T$-provably recursive function is a $\Sigma_1$ formula $\varphi$ with $n+1$ free variables which $T$ proves defines a total function.$^1$ The terms in the language of $T$ itself, as well as their "$\Delta_1$ piecewise combinations," all constitute simple examples of $T$-provably recursive functions. However, in general these account for only a miniscule fragment of the $T$-provably recursive functions. For example, (the usual sequence-based definition of) exponentiation is $\mathsf{PA}$-provably recursive, but is also $\mathsf{PA}$-provably eventually different from every $\{+,\times\}$-term.
I'm interested in whether $\mathsf{Q}$ has any such "surprising" provably recursive functions:
Is there any $\mathsf{Q}$-provably recursive function $\varphi(x_1,...,x_n,y)$ such that for every $\{+,\times\}$-term $t(x_1,...,x_n,z_1,...,z_k)$ we have $$\mathsf{Q}\vdash\forall z_1,...,z_k \exists x\forall x_1, ..., x_n[\bigwedge_{1\le i\le n}x_i>x\implies \neg\varphi(x_1,...,x_n,t(x_1,...,x_n, z_1,...,z_k))]?$$
Intuitively, I'm asking whether there is a $\mathsf{Q}$-provably recursive function which is $\mathsf{Q}$-provably eventually different from every term. Note that "eventually different" here is used in the sense of "different on all inputs with each coordinate sufficiently large." In particular, a function like $$\varphi(x_1,x_2,y)\equiv (y=0\wedge x_1=x_2)\vee(y=1\wedge x_1\not=x_2)$$ is not an example; it is "not eventually the same as" any term, but that's a much weaker condition.
A negative answer would say to me that the class of $\mathsf{Q}$-provably recursive functions is "almost trivial" (which is not to say uninteresting, to be fair).
$^1$Usually we draw a distinction between the function itself and the formula $\varphi$, but since we always wind up talking about $\varphi$ specifically there's nothing lost by focusing on it right from the beginning.