All popular multicolored LED strips these days are WS2812 based, which you could not possibly program without using a MCU.
Hope I've joined the right pedant's thread...
You could use WS2812 sans MCU with some memory and a shift register, perhaps a 74HC165N. Of course, you would need have a clock, and a 555 could be great for that
Well, yes. Implementing the required encoding for the 0 and 1 could be directly stored in the memory.
And for the clock, just a single gate inverter with Schmitt trigger input would do.
Possibly a fun exercise, rather than very practical.
But while this would work with only "discrete" logic, with some programmed memory, technically it would still be "programmable logic". You would have effectively designed a very simple and dedicated processor.
Anyway, the initial question was weird from the start. There are thousands of references of BJTs these days, so is the 2N3904 still a viable transistor?
PIck whatever fits your requirements and (if longevity matters) is reasonably (/or guaranted) likely to be available in a few years from now. Maybe that's called engineering, or something. Not sure anymore.