While thinking of Spy Hunter, I remember there was another game of indefinite length - Xenophobe, also by Sunsoft. I was curious and read the review at gamefaqs and it appears the arcade version was much better. When I was a kid, I didn't know any better; I just thought there was a particular item I could never find (there were many) and that was why the game kept repeating itself. Imagine if you were playing Contra and Konami forgot to put in the end credits; after beating level 8, you would proceed to level 1. As the player, you might assume you did something wrong, because there was no proper ending. As I recall, Contra would cycle indefinitely, but at least there was a sense of finality.
Xenophobe also offered 2 players at the same time. I remember playing that game for many hours with my brother or one of my cousins. There was a general consensus that we were missing some crucial item, so the notion was not peculiar to my mind. I wonder how many other young gamers futilely tried to beat Xenophobe.