Here's a 14 story wood-based apartment building in Norway:

https://www.youtube.com/watch?v=e5XsqauBCX4

http://treetsameie.no/

There are many other wood-based buildings higher than that being planned. The key for most of them is the use of various hybrid wood-laminates.

Of course you're not getting skyscrapers from them but nowadays you can build at least five floors which is not that bad. Most buildings are below that I think.