I am confused about what it is you want to do. You first mentioned an led "matrice", then later you ask about a shift register to display from 0-255. it would be helpful if you could explain EXACTLY how many led's you want to use, and EXACTLY how you want to arrange them. NOTE, a picture is worth 1000 words.

 

IMHO, I think this thread deserves watching.

 

I don't want to strobe the matrix. Surely there is a way to accomplish what I want to do without having to resort to strobing.

Just drive the LEDs directly. Strobing is used to decrease circuit complexity. You don't have to use it.

If you wanted to drive an 8x8 matrix without multiplexing, you'd need up to 64 drivers. If you allowed multiple LEDs to be on at the same time (e.g. an entire row or column), you'd need 9 drivers.

 

electron beam point scanning?
I've considered charlieplexing. It's fine for smaller arrays, but, according to my research, it's more complicated with bigger arrays. It's necessary to switch the pins back & forth from input to output. That complicates the code. Perhaps a serial in/parallel out shift register. What's an external refresh drive? Is that different from an LED driver? Is serial LED the same as LEDs in series? I wonder about SPI or I2C. Please elaborate on "segments as phases inversed".

when you cycle through all 7 segments instead of the digits. then put a 4-bit pattern to light up the appropiate digits.

less brightness variation.