The answer will depend on which airline you are flying, and possibly what fare you have purchased. Some airline change fees are per ticket, some are per direction, some are per segment, and I'm sure airlines have found still other ways of assessing them.
Most airlines charge a change fee per ticket per passenger without regard to the number of segments in the trip. That is, if you have a single reservation for two passengers on a round trip with one connection in each direction (2 segments per passenger per direction), you'd pay two change fees to leave a day later (plus fare difference).
Charging per segment per passenger is a feature I associate with discount carriers such as Allegiant or Vision, but other airlines in that category do not, like Spirit or EasyJet.