I love this, but I have a couple thoughts on the relationship between the client and service.

I think there may be some ways we can improve the relationship between the client and service here. Right now, HostingService is either given a client, or will construct one by itself, and having an account requires doing a subclass and constructing it and passing it to the HostingService's constructor.
I'd propose we change this design a bit to ensure client construction happens in a more consistent way. I'd suggest doing this by having a build_client method on HostingService that constructs a HostingServiceClient by default, passing in *args, **kwargs to it. HostingService.__init__ would set self.client to this, passing service=self as a parameter (allowing any client to access the parent service and therefore account).
GitHubClient would then just override build_client to return an instance of GitHubClient instead. It wouldn't have to do anything special with the account, and would just pass the arguments it's supposed to pass.
This makes it easier for us to add additional default parameters down the road and to keep HostingService subclasses clean.

Missing trailing period.
It'd also be nice to flesh this out more and say what its responsibilities are, and that it can be subclassed for the same reasons you outline in the review request's description.

While we're here, can we add some docstrings to these methods?

With the above, this would just be:
self.client = self.build_client(service=self)

Ship It!