My supervisor at work gave me a UPS with a weak battery. It posts with the battery in, but if you try to load it more than a few watts it quits.
It the trapezoidal shape APC model BP500UC. This model has been discontinued and the batteries cost almost as much the the whole thing, so they just toss them out at work when they quit. I had to buy the team leader a similar UPS for her computer because the owner is too stupid to protect his equipment. We had a 2 second power blip last month when someone hit a pole down the street and Eleanor's computer crashed. The one I use still has a functioning UPS. Most of the computers in the building are on a central backup and protected outlets, but not all the building is wired that way.
This one has a test button on the front so you can force it on and use it off-grid. It's available for the cost of shipping anywhere in the US. Figure about $12. Battery not included, it's going back for recycle.