Find the maximum $y$-coordinate of a point on the graph of $r = \sin 2 \theta.$
For $r = \sin 2 \theta,$
\begin{align*}
y &= r \sin \theta \\
&= \sin 2 \theta \sin \theta \\
&= 2 \sin^2 \theta \cos \theta \\
&= 2 (1 - \cos^2 \theta) \cos \theta.
\end{align*}Let $k = \cos \theta.$  Then $y = 2 (1 - k^2) k,$ and
\[y^2 = 4k^2 (1 - k^2)^2 = 4k^2 (1 - k^2)(1 - k^2).\]By AM-GM,
\[2k^2 (1 - k^2)(1 - k^2) \le \left( \frac{(2k^2) + (1 - k^2) + (1 - k^2)}{3} \right)^3 = \frac{8}{27},\]so
\[y^2 \le \frac{16}{27}.\]Hence,
\[|y| \le \sqrt{\frac{16}{27}} = \frac{4 \sqrt{3}}{9}.\]We get $y = \boxed{\frac{4 \sqrt{3}}{9}}$ when $k^2 = \cos^2 \theta = \frac{1}{3},$ so this is the maximum $y$-coordinate.

[asy]
unitsize(3 cm);

pair moo (real t) {
  real r = sin(2*t);
  return (r*cos(t), r*sin(t));
}

path foo = moo(0);
real t;

for (t = 0; t <= 2*pi + 0.01; t = t + 0.01) {
  foo = foo--moo(t);
}

draw(foo,red);

draw((-1,0)--(1,0));
draw((0,-1)--(0,1));
draw((-1,4*sqrt(3)/9)--(1,4*sqrt(3)/9),blue);

label("$r = \sin 2 \theta$", (1.2,0.6), red);
label("$y = \frac{4 \sqrt{3}}{9}$", (-1, 4*sqrt(3)/9), W, blue);
[/asy]