The undecidability of Hilbert's tenth problem implies the following (there is a stronger statement here, Theorem 9):

For any computable function $f$, there is a family of integer polynomials (where some coefficients are allowed to vary) which have integer solutions, but the growth of the least integer solution (in absolute value) with respect to the varying coefficients is not bounded by $f$.

Hilbert's tenth problem over the rationals is not known to be undecidable. However, even if it were, the equivalent rational statement would not follow, since one could grow the denominator to get "inaccessible" solutions rather than the absolute value.

Nevertheless, I'm curious whether some kind of superpolynomial growth (e.g. exponential) is known to be achievable (or unachievable). So, for example, is there a system of equations of the form $$P(x_1,\ldots,x_n,\beta_1,\ldots,\beta_m)$$ such that for every rational $\vec\beta$ there is a rational $\vec x$ satisfying them, but the length of the smallest rational solution vector for a given fixed $\vec \beta$ is not bounded by a polynomial in the $|\beta_i|$? Or, conversely, is there some reason that this would be impossible?