Please read the article:

A. Tyszka, A hypothetical way to compute an upper bound for the heights
of solutions of a Diophantine equation with a finite number of solutions,
Proceedings of the 2015 Federated Conference on Computer Science and
Information Systems (eds. M. Ganzha, L. Maciaszek, M. Paprzycki); Annals
of Computer Science and Information Systems, vol. 5, 709-716, IEEE Computer
Society Press, 2015,

http://dx.doi.org/10.15439/2015F41

Please read the preprint (updated on January 2017):

A. Tyszka, A conjecture which implies that there exists a computable upper
bound for the heights of solutions of a Diophantine equation with a finite
number of solutions,

https://arxiv.org/abs/1109.3826

Let $f(1)=2$, $f(2)=4$, and $f(n+1)=f(n)!$ for every integer $n \geq 2$. We conjecture that
if a system $S \subseteq \{x_i \cdot x_j=x_k, x_i+1=x_k: i,j,k \in \{1,...,n\}\}$ has only
finitely many solutions in positive integers $x_1,...,x_n$, then each such solution
$(x_1,...,x_n)$ satisfies $x_1,...,x_n \leq f(2n)$.

The conjecture implies that there exists an algorithm which takes as input a Diophantine equation,
returns an integer, and this integer is greater than the heights of integer (non-negative integer,
positive integer, rational) solutions, if the solution set is finite.

extremelysurprised if something like that were known to hold for a wide class of sets. You can assume without loss of generality that every value is attainedinfinitelymany times, and since usual proofs of the MRDP theorem proceed by throwing in auxiliary variables all along the way, it’s almost inevitable they end up like that even without asking. $\endgroup$ – Emil Jeřábek Nov 10 '16 at 15:48