Find the largest value of $x$ for which
\[x^2 + y^2 = x + y\]has a solution, if $x$ and $y$ are real.
Completing the square in $x$ and $y,$ we get
\[\left( x - \frac{1}{2} \right)^2 + \left( y - \frac{1}{2} \right)^2 = \frac{1}{2}.\]This represents the equation of the circle with center $\left( \frac{1}{2}, \frac{1}{2} \right)$ and radius $\frac{1}{\sqrt{2}}.$
[asy]
unitsize(2 cm);

draw(Circle((0,0),1));
draw((0,0)--(1,0));

label("$\frac{1}{\sqrt{2}}$", (1/2,0), S);

dot("$(\frac{1}{2},\frac{1}{2})$", (0,0), N);
dot((1,0));
[/asy]
Hence, the largest possible value of $x$ is $\frac{1}{2} + \frac{1}{\sqrt{2}} = \boxed{\frac{1 + \sqrt{2}}{2}}.$