Screen capture at more than 1000 FPS
I can't rule out specialized hardware devices that might be able to, but 1080p video at 1000fps produces 6GBps of raw data at 8 bit color.  That's 48 gbps, which exceeds the highest standards of most current connectors.  Display port 1.4 is just shy of 26gbps and barely covers half of the needed data rate for a 1000fps 1080p data stream.

You might be able to pull it off with a quad-link 12G-SDI setup, but I'm not aware of any hardware that can work with quad link 12G-SDI.

As far as handling it internally, no modern single SSD can handle the raw data rate.  An ultra top-end PCI express SSD can still only write around 2GBps, so you'd need over 3 of them working in parallel to keep up with the raw data rate.  At PCI express 3 speeds, you'd need 7 lanes minimum for the data transfer alone, so you could theoretically handle it with a PCI-E 3.0 x8 card, but you also need to take in to account the data that needs to be sent to the card for rendering at that rate as well.

You might be able to reduce this a bit by processing in CUDA on the card itself, but that's still an extremely high amount of processing to be doing while also trying to render whatever it is you are rendering at 1000fps, which, btw, you can't even display at 1000fps because there's no way to get a 1000fps feed out of the computer or display it.

Overall, I suspect the reason you don't find any software to do this is that it simply isn't possible or practical to do at this time.  Limits are probably set based on what seems like rational limitations for the capabilities of current hardware.