What is the maximum height a settlement can reach in Fallout 4?

Yes there is and it differs between each settlement. The settlement boundaries are a box extending upwards towards the skybox but the ground height varies between settlements.

Crediting the above link, here are some examples;

• Sanctuary - 6 stories
• Tenpines - 3 stories (From the highest point)
• Starlight - 9 stories
• Greentop Nursery - 7 stories (From the level of the house)

...where "stories" is defined as a wall with a roof (with the next story being another wall on top with another roof, etc)

20 stories at Abernathy farm. I play on Xbox one. I used the reset settlement size glitch.

Anything above the fifth floor kicks you out of build mode in Sanctuary. Also, I've noticed that even using the weapons trick to add more settlement size to the meter is okay to do but by the end of the second go round on using the trick, the graphics start getting a little twitchy. Nothing major but I wouldn't suggest pushing past using the trick more than twice but that is filling up the meter, clearing it completely and repeating the process once the meter fills up again. This will not add to your build height as that as set by a different limit. However, you can build one massive facility. So, long as you take it easy on the decorations and don't waste time with clutter, you will be able to make a serious fortress.