I haven't noticed any obvious bound on the size of the solutions of a Diophantine equation.
There may be some results for quadratics such as $ax^2+by^2+cz^2=0$ has a solution bounded by $abc$ (that is probably wrong but I read something along that lines?).
- What results are there for small degrees, small numbers of variables?
- What examples of there of huge numbers defined by small Diophantine equations?
edit: Related Does the equation $x^4+y^4+1 = z^2$ have a non-trivial solution? but I haven't read it yet.