Capacitor in a fluorescent lamp starter
The capacitor in old Fluorescent Starters is for EMI suppression.  This is typically a fairly-small value - anywhere between 1n to 100n, depending upon who made your particular starter.

The capacitor may also reduce contact erosion on the starter contacts - I honestly don't know.  But I do know that in olden days when everyone had an AM radio sitting on the kitchen counter, you could immediately tell if someone turned on a Fluorescent lamp that didn't have that capacitor inside the starter.