The copy of String of Disappearances has the ability to copy itself. The string can go on or not, as long as players are able and willing to keep paying {U}{U}.

While resolving String of Disappearances, you can’t pay {U}{U} multiple times to copy it multiple times. It’s a string, not a web.

The copies String of Disappearances creates are created on the stack, so they’re not cast. Abilities that trigger when a player casts a spell won’t trigger.

If the target creature is an illegal target by the time String of Disappearances tries to resolve, the spell doesn’t resolve. That creature’s controller can’t pay to copy String of Disappearances.