Gotta have an ancient bow for those long shots you can't afford to miss, as well as to make it super easy to farm the specific dragon part you want/need ...
Gotta have a few Savage Lynel bows - 3 or 5 shot versions - and hopefully with the quick-shot or durability buff. I don't need the attack buff since they are so ridiculously powerful to begin with.
And, I just love the Great Eagle bow. It's just, great. And, being wood makes it perfect for use in a lightning storm if you need it, so that you don't use up durability on your ancient bow.
The Duplex bow is a good one to have with you as well, since it truly does double-damage with elementals, where (I think) most of the other bows only count one arrow's elemental effect, even if multiple arrows hit.
Nothing wrong with carrying some single-shot bows though ... Royal Bows and the steel Lizal bows are worth holding onto if you have an open slot, but I'm never going to pass on one of the above bows for one of these unless I know one of the above is about to break ... then I toss it, grab the royal/lizal bow for the time being, but with the intention of replacing as soon as I can with a multi-shot bow.