Early Mario games were much more punishing than most modern titles. Mario, who would die instantly from any damage, was given a number of lives that he could expend without being set back too far. Once those lives were used up, however, the entire game restarted from level one.
Expending all of Mario's lives triggers an event referred to as a fail-state: The point where the player has officially failed at their given task and must start over.
Modern day games are quite different from retro platformers. Super Meat Boy has infinite lives—if put in the context of Mario's design—and therefor Meat Boy's only fail-state is death, and the only set back is to the beginning of the current stage.
It's often very difficult to find a game without a fail-state that still manages to challenge the player. We can, however, invent one. Imagine a version of duck hunt that will spawn ducks forever, and ends when the player manages to shoot thirty ducks in a row without missing. When the player takes out a duck, a counter ticks up, but when the player misses, the counter resets. This game has no fail state, but only because there would be no point in rolling the game back. In a since, resetting the counter is as much a fail state as respawning in Super Meat Boy. The only true way to completely remove any simbalince of a fail state is to never reset the counter, but, at that point, the player has virtually no reason to exert themselves. They will, eventually, take out thirty ducks, no matter how lazily they try.