It's of course totally possible for one to make a client that reads responses regardless of whether it send a request, but that's rather silly as giving up flow control like that immediately opens your application up to a DoS attack. (Of course just because it's silly doesn't mean no-one does it!)
I wonder if that's really how browers work or if they employ an array of open connections that are periodically polled for responses to outstanding requests.
Obviously, responding to something that wasn't requested is a bad idea.
Flow control only works for large responses (which is good, because that is at least one resource that you can protect), makes you wonder what you could do with multiple answers small enough to fit in the same window and if that would allow you to identify HTTP implementations that have taken 'asynchronous' one step too far.
So, presumably if someone requests anything from your site you can keep bombarding their browser with unrequested content that will get queued. As jacquesm indicates having an array of connections with reserved queues would avoid this blocking requested content.
So effectively all that can happen is a website can DoS the connection to itself.
(Yes, an attacker can try to initiate many many TCP connections. This uses much fewer resources than an actual HTTP session would but can still be an effective DoS attack. This is known as a SYN flood.)