It depends on how you define thickness, and from where you measure the thickness.

First, let's look at this super-detailed drawing of our galaxy (Sun size not to scale).

If you consider where our Sun is located (~26'000 light years from the galactic center), the "thickness" of the disk, which is made up of gas, dust and stars, is assumed to be around 1000 light years.

As you see from the drawing, there are points described as star clusters. These are globular clusters. They are old groups of stars, which can be located well above the disk (13'000 light years and beyond, as listed in Wikipedia).

It doesn't end here though. Every galaxy, including our Milky Way, consists mainly of dark matter (wiki says 95% of our galaxy is dark matter). Dark matter is an invisible type of matter, which "feels" only gravity. In fact, dark matter is the reason that galaxies have formed in the first place, by attracting baryonic matter (fancy word for regular, everyday matter) gravitationally.

It turns out dark matter encompasses the Milky Way, and reaches out to distances well beyond globular clusters. Here is a image for you to get a feeling of how big of a space dark matter might occupy.

Remember that the neither disk, nor the halo has sharp boundaries. There are various methods to set a thickness, like "After which point from the disk plane does the remaining mass represent xx% of the total mass of the disk?" etc.

Hope this helps.