And it’s getting annoying! Of course I can’t argue with a fax machine to tell it that I’m not a fax machine. And I found some sites to do a reverse lookup to find out where it is, but they all want money to give me any useful information. Argh!
If you have a fax machine you could always fax something back saying stop calling this number, it’s not a fax. Course then they’ll have your real fax number.
Why do you need to do a reverse look-up? Can’t you just fax them back?
Are you in the US? Try AnyWho reverse lookup.
Or…
Have you got a fax machine you can connect to that line? If you do, and the fax comes through, look for a contact phone number on it.
If there’s no contact number, fax them back and tell them they’ve got a wrong number.
ETA: Double Ninja’d!
Faxing them back is a good idea, but I have to find a fax machine! I think there’s one upstairs.
To clarify “connecting” to the fax machine: Transfer the incoming fax call to your in-house fax machine’s number. No need to plug a machine into your phone’s line. 
It was easier just to fax them from a different machine. Let’s see if this works!
Sometimes in my company, they change one of the office’s fax numbers but don’t communicate it to the employees. So we keep faxing stuff by email (using fax2mail) to a wrong number, sometimes for days, weeks, or even months. If your phone number is a recycled fax number from someone else, this might be why it’s happening. You might not have a lot of success stopping it (depending how popular the number used to be), but I agree that sending a fax to the fax number is the best way to try.
After I sent the fax, I had 2 more calls, but they seem to have stopped. We’re approaching about 45 minutes of no calls.
I’m pretty sure it’s not a recently recycled number, as I’ve had it for over 9 years now.
This happens to me at work fairly often, probably once every couple of months or so, and I’ve never been able to figure out why. If I keep picking the phone up when it rings the fax machine will continue to try to call me back, but if I let it roll to voice mail then it usually stops. The last time this happened to me I googled the number and it turned out to be the financial aid office at a university in Washington state. So I just sent them an email telling them their fax machine kept trying to call me and they sent me back a reply the next day telling me it had been corrected.
Speak to both machines separately and explain that corporate policy doesn’t forbid these office relationships but they can’t be allowed to interfere with company business like this.
Be sure to have an impartial third party, like the pencil sharpener, in the room as a witness.
I feel your pain. We got a new phone system in June. About once a week I get phone calls from a fax machine. The machine is calling the back line which is a different extension but it’s still a pain in the ass.
I just pick up and hang up.