The only real answer we can give to this is it depends
It depends on the airline, the nature of the legs (long/short), the specific airports/countries involved (re-fuelling costs can vary significantly) and route profiles used.
If the aircraft is doing 4 very short legs, the cost of carrying the fuel may be less than the cost of refuelling three times. Equally it's certainly faster to fill the plane once, rather than 1/4 fill it 4 times, because so much time is spent in setting up etc, and you can't usually re-fuel with passengers on board.
On the other hand, if legs 1 and 3 are nearly at the endurance of the aircraft, you physically can't carry fuel for legs 2 and 4: your fuel tanks aren't big enough. Equally on longer routes, the cost of carrying extra fuel increases dramatically.
Flip that around and say that legs 1 and 3 are short, but legs 2 and 4 are long (without being at the aircraft's endurance) and you have a problem that during legs 1 and 3 you're carrying a lot of un-neccessary fuel.
All airlines will have different policies and procedures, and those policies will vary wildly depending on the fuel costs at different airports. But I'll try to summarise at least some of the options here
- For lots of short routes, fuelling once for multiple legs can be viable, but may not be the best option depending on fuel costs at each location. You wouldn't necessarily do a full day on one re-fuelling instance, but you may fuel twice for 4 legs, for example
- For a short leg followed by a long one, it's rarer but the above will still reply
- For a long leg followed by a short one, it's unlikely to be cost effective, although in certain situations your second leg's fuel load could be close to the reserve needed for the first leg: as long as you don't have to eat into your reserve on the first leg, you don't necessarily need to refuel.
- For multiple long legs, you almost certainly don't have the fuel capacity, and carrying so much extra fuel for so long makes little sense: that's expensive.
In short, you usually want to fly with as little fuel as possible - a light aircraft is cheaper to run, can hold more cargo/luggage etc.
In a few situations, the cost of refuelling can make it worthwhile to fuel up in one location, where the cost of fuelling there makes up for the cost of carrying that extra fuel around. That's a niche scenario though, and wouldn't be the "usual" mode of operation for most airlines