Enough to fill your reel nearly to the top,its kinder to your line,the loops of line are more open and bigger,so not as prone to memory,ie stopping in coils when you pull it of the reel,and if you hook a big fish you have more line to play with.
And it depends on the type of backing. Most reels sized for a certain line, say a weight 7 will take about 70 to 80 yards of backing.
For most of the fish you catch, especially in rivers you will never get into your backing even with quite large fish.
However I have had quite modestly sized rainbow trout, in the 3 to 4 pounds bracket, run out well over 100 yards of backing in shallow water. This sort of activity by the trout will often astound those who are used to catching coarse fish, even large carp and barbel.
A fit overwintered rainbow trout in prime condition is one of the hardest fighting species size for size in the world. Especially if caught from a reservoir like Rutland or Grafham. The hardest fighting rainbow trout I ever caught weighed 41/4 lbs and was taken from a 300 acre lake at 6500 feet above sea level from a boat.
It was an incredible scrap. I'll never forget it. It took over 150 metres of flattened nylon backing from my spool, plus 10 metres of shooting head.
If you have two spools for the one reel then you can do a little trick.
Attach the end of the fly line (where you would attach the leader) to the reel and wind it all on, then attach the backing to the fly line and wind it all on until the reel is almost full up. Then you can attach the end of the backing to the second spool and transfer it across. You will then be presented with a full spool with the backing and fly line in the correct positions.