Compute $\arctan 1 + \arctan \frac{1}{2} + \arctan \frac{1}{3}.$  Express your answer in radians.
Since $\tan \frac{\pi}{4} = 1,$ we know that $\arctan 1 = \frac{\pi}{4}.$

Now, consider the triangle in the $2 \times 3$ grid below.

[asy]
unitsize(2 cm);

filldraw(arc((3,1),0.5,180 - aTan(1/3),180)--(3,1)--cycle,paleblue,white);
filldraw(arc((3,1),0.5,180,180 + aTan(1/2))--(3,1)--cycle,palered,white);
draw((0,0)--(3,0));
draw((0,1)--(3,1));
draw((0,2)--(3,2));
draw((0,0)--(0,2));
draw((1,0)--(1,2));
draw((2,0)--(2,2));
draw((3,0)--(3,2));
draw((0,2)--(1,0)--(3,1)--cycle);
[/asy]

The red angle is equal to $\arctan \frac{1}{2},$ and the blue angle is equal to $\arctan \frac{1}{3}.$  Furthermore, the sides of the triangle are $\sqrt{5},$ $\sqrt{5},$ and $\sqrt{10},$ so the triangle is a $45^\circ$-$45^\circ$-$90^\circ$ triangle.  Therefore,
\[\arctan 1 + \arctan \frac{1}{2} + \arctan \frac{1}{3} = \frac{\pi}{4} + \frac{\pi}{4} = \boxed{\frac{\pi}{2}}.\]