This is the use case:
"...5 doughnuts to be awarded each consecutive day you visit the shop until the ???? is broken."
I'm thinking it's going to be along the lines of consecusion or consecutivity....??? But these are of course not real words.
For more clarity: I don't want to have the term bound to any misleading things. The use case I gave was an example but this term needs to be abstracted due to the nature of the real problem as it relates to software. i.e. String, chain, etc.