Somewhere (discussed on HN) someone devised a "better-than-perfect" compressor. Most inputs get compressed (smaller than input), except for one input that does not. That one input is cryptographically impossible to find - or something along those lines.

Unfortunately I can't find the article I'm describing here, maybe someone else can? It was a long time ago so I might be misrepresenting it slightly.

It can't exist.

Presume this compressor guarantees the output will always be at least one byte smaller (with the exception of the impossible to find input)

So just keep running your data in a loop through the compressor. If you start with a 1MB file, it will take a maximum of a million iterations until the output shrinks down to zero bytes, which is the smallest possible file. Record how many iterations it too.

You can now extract your file by feeding a zero byte file into the decompressor and running the same number of iterations. Which You can now store every 1MB (or smaller) file in the world in just 20 bits.... But that would means there are only 1 million possible 1MB files?

Even if you put some minimum output size limitation on the compressor, say it can't produce any file less than 512 bits, the same argument applies. It's just that the numbers get bigger.

That's how all compressors work, in that likely files (eg. ASCII, obvious patterns, etc) become smaller and unlikely files become bigger.

You could design the opposite, a system where ultra-rare files get compressed a lot but most don't compress.

But as stated, what you said is not possible. To compress by a single byte, only 1/256 files can be compressible, or you'd have multiple different inputs compressing down to the same output.