How deep is a rainbow?

You don't really "look at the rainbow". You look at light that was refracted in rain drops. Where the drops are, the refraction happens. And as long as the density of water drops does not impede your view, the depth is "infinite".

In other words - any drops that present a certain viewing angle (in the triangle sun-drop-eye, the angle subtended from sun to eye is the same) will give the same color, regardless of the distance.

So there is no real answer, because it will depend on the density of the rain drops. Once the probability of a drop being in the way becomes substantial, you reach the limit of your rainbow "depth".

"Heavy rainfall" is considered anything above 0.3" (7.6 mm) per hour. Rain drops fall at rates between 1 m/s and 10 m/s (source). Let's say the average size is 2 mm with a terminal velocity of 6 m/s; This means that in 1 hour, a column of about 20 km (6*3600) "falls", creating a layer of water that is 8 mm thick. When a rain drop (volume $\frac43 \pi r^3$) hits the ground (area $\pi r^2$ it creates a "puddle" of depth $\frac43 r \approx$ 1.3 mm deep. Six of these drops would correspond to 8 mm, so on average mean free path between drops is 20 km / 6 = 3 km.

That means that in "heavy rain" you expect to see a rainbow with a depth up to 3 km. As the rain gets heavier, the depth reduces; if it's lighter, it can be more.

The above is very approximate...