Is this possible?
Say you farmed your computer chip manufacturing to where it was cheap, and when you received a shipment you ran a few through the tester and they worked fine.
Could there be an entire layer of circuitry, essentially another chip piggy-backed on that chip, that could be used to, oh, say, give an outside power the ability to ‘listen in’ and monitor or to take over any machine using that chip?
Sure, in a Terminator movie.
You’ve been watching too many Sci-Fi movies.
First off you have to understand that the chips used are NOT your standard chip you find in a PC…Far less sophisticated…The programming in actually pretty simple…All it basically does is look at the readings from all the inputs and then looks them up in a table…and the table tells them what adjustments it needs to make. This simple processing is done several times a second.
The re-programming of one of these chips…actually isn’t programming at all…It just changes the values in this table.
As for Testing Chips…I’d be surprised if each and every chip isn’t tested before it goes in a car. Once the test is devised. The hardest part is designing and programming the test.
say, give an outside power the ability to ‘listen in’ and monitor or to take over any machine using that chip
This is where I really hate TV. Sorry…but what you THINK may happen is IMPOSSIBLE…Without getting into basic computer architecture…A chip can only work in conjunction with a Circuit board. If that circuit board isn’t connected to any outside device…there is no way the chip can make any connection (or control).
Possible, I would guess so. Practical no way, the cost would be far too high for very little if any gain.
Take you meds and be happy. 
What kind of chip are we talking about? If you’re talking about your home computer, it’s possible I suppose, but the problem is that it would have been detected almost immediately after being sold. There’s all kinds of software that will show you the network traffic to/from your computer. (Wireshark among others) This network traffic must meet standards or the computer wouldn’t be able to talk to anything. Any traffic that doesn’t meet standards wouldn’t make it past your home router or internet connection. Any packets that were going to strange places would be noticed by the “geek community” (of which I am a member, I suppose) almost instantly. Same goes with any weird radio signals. There’s another benefit of still living in a semi-free country
I suppose as an addendum, I should mention that there’s all kinds of viruses and other malware that do exactly what you describe, for the purpose of harvesting passwords, credit card numbers, etc. Some are employed legitimately and illegally by law enforcement and/or spouses, PIs, etc. to snoop on chat sessions, emails, etc. Most good Antivirus software (if working properly and up to date) will detect the vast majority of all but the most sophisticated efforts here.
Yes, it is possible. However, you do realize that it would require more than just a hardware modification, right? There are layers of “coding” surrounding the hardware, much like the layers of an onion. In order to access your little chip master, you would need to have knothole(s) in the upper to lower layers of control code to your hijack hardware. The micro doesn’t know anything about the stuff “above” it, to put it simply.
Don’t think this hasn’t been thought of before and measures taken to insure that it doesn’t happen on sensitive hardware. There are many reasons why advanced technology microprocessors come under the auspices of ITAR control…
I assumed the OP was talking about a chip on a car (since this is a car forum)…but if he’s NOT…yet it’s possible…but NOT likely. Before you buy a computer the chips on the computer tested by the board manufacturer, the integrator…It would have been detected…
HOWEVER…Back during the First Gulf War…The CIA intercepted a printer from that was being shipped to Iraq from Jordan. Somehow they knew this printer was to be used by the Iraq military. A special chip was designed and put in the printer that when hooked up was able to access Iraq’s Defense network…And SHUT down certain pieces of their defense system.
http://securitydigest.org/virus/mirror/www.phreak.org-virus_l/1992/vlnl05.007
Those F-117’s and the Apache’s with hellfire missle’s were also very effective at shutting down Iraq’s defense systems. The “infected chip” method does have a certain elegance to it.