I know that Kronecker claimed it was God's doing, and that even prehistoric humans used some ways of counting. But I am curious where the idea of a sequence of numbers stretching out into infinity appears for the first time explicitly. I suppose another way to phrase it would be to ask who invented infinity. "Ancient cultures had various ideas about the nature of infinity" is what Wikipedia says.
Pythagoreans must have understood it already, and Aristotle even discusses the difference between potential and actual infinities, so it had to be before that. Maybe they borrowed it from Egyptians? Babylonians had a positional system which allows recording arbitrarily large numbers in principle. But is that enough? What gives me pause is the history of zero. Babylonians and Alexandrian astronomers were using it as a placeholder for centuries before the concept of zero as representing nothing was formed, and not by them. And if it happened to zero it could happen to infinity. By the way, how ironic that infinity was discovered before zero.