You sue them. Depending on the circumstances, you may be able to do so in small claims court, which is pretty inexpensive to do. Google cannot bring their usual insane level of legal resources to bear in this environment, and you will most likely win.
I'm a little confused here - you sue them, in small claims court, for not providing a service address?
How do you serve them without an address, for a claim of not providing an address?
I mean, I know what happens when I appear in court as the applicant/plaintiff, and tell the judge that the respondent/defendant was never served.
Okay, lets say the judge doesn't laugh my case out at that - how is my claim going to proceed? "I claim $X monies for not supplying me with their service address"?
(My understanding is that, by not providing a service address, all you can do is report them to specific bureaus, arbitration councils, etc. You cannot sue businesses, as far as I know, for not having or not providing a service address).
AFAIK them not providing an address is not the main point, it's them not collaborating on your case for a copyright infringement. The not providing an address is just more evidence of them not following on their responsibilities regarding the copyright infringement.
If you are in the US you can look up their official service address through the secretary of state for the state that you will be filing the lawsuit in. There is never a need for them to specifically provide you an official address.
And if you're filing a state case, and you know which county they are in, call the Sheriff. Every Sheriff's office deals with legal service on a routine basis. I've paid the Sheriff many times for serving complaints.
It seems to me that without know that, you're still making empty threats, only doing it passive aggressively.