Using the Brewbit ChibiOS/RT branch did not solve my problem, but I figured out where those extra fields on the thread struct came from:
There is a "#define THREAD_EXT_FIELDS" in chconf.h which can be used to add fields. Since I am trying to use the brewbit CC3000 driver in my own project, I used my own chconf.h and I missed to add this define. And you also have to add the '#define THREAD_EXT_INIT_HOOK' from brewbit's chconf.h to get those extra fields initialized.
Now I can get my CC3000, which is connected to a STM32F103CB (128 kb Flash, 20 kb Ram) up and running using the brewbit driver (using ChibiOS 2.6.3 FWIW). I can connect to an AP and DHCP works and I can ping my board. But when I tried to send a package from my board to a computer on my internal network (using sendto(...)) I hit an error.
First I thought this was some kind of memory overflow and therefore I tried my code on a different board with a STM32F103RG (1Mb Flash, 128 kb Ram), but I still get the same error/lockup. I guess I need to do some more debugging...