For $k\in{\mathbb Z},k\neq 0$, denote by $f(k)$ the number of integral points on the Mordell curve $y^2-x^3=k$. According to the data at http://tnt.math.se.tmu.ac.jp/simath/MORDELL , the largest value of $f$ on the interval $[-10000,10000]$ is 32, attained for $k=1025$.
Is it known/conjectured whether $f$ can take arbitrarily high values ?
The largest value for $f(k)$ that I know of is an example due to Noam Elkies with $k = 509142596247656696242225$, where there are (at least) $125$ pairs of solutions (so $f(k)=250$ in your notation). If ranks of elliptic curves over the rationals are absolutely bounded, then so is $f(k)$ (as long as one restricts to $6$th power free values of $k$ to avoid trivial scaling). Nothing is provable at this point, however.