Is it possible to 'fork a fork' in Github?

Simple answer: Yes and no.

No, for a single account as you are unable to create two repositories with the same name (and forks always have the same name).

Yes, for multiple accounts. So you could setup an organization for your second fork.

This is now very possible. You just go to the fork you want to fork and click "fork" like you would always do. This only works if you don't have a different fork of that repository in your account already, though. If you do, you have to delete it.

Answered in a comment by iFreilicht

Since this is the correct answer now, it is better to highlight it.

I contacted GitHub support about this 'issue' and they confirmed this is not possible. Also, it is very unlikely that such a feature will be implemented in the future.