No, the blinds never go up in a cash game. The amount of the blinds is how you know you’re in the right game for your budget, or how you know how much cash you should be prepared to loose.
But maybe you don’t really understand the nature of cash games. “Extend the night indefinitely” is exactly what happens - if you don’t pick a time to stop, like 1:00 AM, you might end up playing all night, and lots of people are good with that. It isn’t like a tournament, where somebody wins the tournament by winning all the chips. It’s more like, play for a while, see how much you can win.