Examples of Diophantine equations with a large finite number of solutions I wonder, if there are examples of Diophantine equations (or systems of such equations) with integer coefficients fitting on a few lines that have been proven to have a finite, but really huge number of solutions? 
Are there ones with so large number of solutions that we cannot write any explicit upper bound for this number using Conway chained arrow notation?
Update: I am also interested in equations with few solutions but where a value in a solution is very large itself.

Let $b$ is a non-zero integer, and let $n$ is a positive integer.
The equation $y(x-b)=x^n$ has only finitely many integer solutions.
The first solution is:
$x=b+b^n$ and $y=(1+b^{n-1})^n$.
The second solution is:
$x=b-b^n$ and $y=-(1-b^{n-1})^n$,
cf. [1, page 7, Theorem 9] and [2, page 709, Theorem 2].
The number of integer solutions to $(y(x-2)-x^{\textstyle 2^n})^2+(x^{\textstyle 2^n}-s^2-t^2-u^2)^2=0$
grows quickly with $n$, see [3].
References
[1] A. Tyszka, A conjecture on rational arithmetic
which allows us to compute an upper bound for the
heights of rational solutions of a Diophantine
equation with a finite number of solutions,
http://arxiv.org/abs/1511.06689
[2] 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.
[3] A. Tyszka, On systems of Diophantine equations with a large number of integer solutions,
http://arxiv.org/abs/1511.04004
