Inspired by Pi Han Goh

This is a partial solution.

There is a nice pattern for integers $x$ and $y$ so that $\sqrt{x}-y \approx 0.888$ .

The pattern is as follows: $x_1=79$ , $x_n = \left\lceil (\sqrt{x_{n-1}}+9)^2 \right\rceil$ for all $n\ge 2$ .

$y_n = 9n-1$ .

After @Calvin Lin's comment: Here is a better way to understand: $x_n= \left\lfloor \left(9n-\frac{1}{9}\right)^2 \right\rfloor$ .

From the table below, the next value of $x$ after 320 is 723, and the corresponding $y$ is 26. So $x+y=723+26=749$ .

(This solution is yet completed as we have not proved that 749 is the minimum value of $x+y$ .)

Now you may want to prove that $\lim_{n \to \infty} (\sqrt{x_n}-y_n) = \frac{8}{9}=0.888888888888...$