My first dedicated gaming system was the PS1, and in general I have no trouble going back to the sprites and chunky polygons of the mid-to-late 90s, whether on consoles or computers
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as long as I can upscale them to 1080P
Beyond that it gets a bit hit and miss- SNES and Mega Drive games look and sound fine to me and I’ve played plenty of 16 bit console games as an adult. On PC, I can enjoy 2D stuff like Sam & Max Hit the Road or the original X-Com but most early 3D, like the original System Shock, looks a bit too much like visual vomit.
Going to 8-bit, while the vast majority of NES games are too primitive to my eyes and ears, I have no problems with Game Boy/Game Boy Color games.
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(Well, at least the good ones, mostly made by Nintendo)
Is it just nostalgia because I had a GBC as a kid or is it because Game Boy games came later and had more developed visual aesthetics? 
My limit is probably the very late 80s
It depends what kind of game - plattformers, casual games? Can get as old as it wants.
For more complex games like RPGs, probably the early 90s is as old as it gets, otherwise you reach a point where many no-brainer quality of life features disappear, such as saving your game.
The oldest FPS I played to completion was Star Wars: Dark Forces (1995), and it’s actually a really fun game… once you get used to the antiquated controls. The late 90s in general are a time where there were a lot of good and increasingly complex games released without being too much of a technological shock for modern people looking back at old games.