What would be a realistic number of arrows for an archer to have/carry/bring with him? Dozens or hundreds?
If we're talking about how many arrows an archer can carry, I'd say it's limited only by carrying capacity - if you want to fill your backpack up with arrows, good for you.
I suspect what you want is: how many arrows can an archer fire in one encounter? (Keeping in mind that getting an arrow from a quiver is essentially "free", as opposed to retrieving something from a pack). At that point, I'd probably rule you're limited to two quivers - one on the hip, one on the back. If you don't have a backpack, I'd probably let you have a third (on the opposite shoulder). Once you've exhausted those, then I'd say you'd have to take actions to retrieve another quiver from your pack.
Going by the pathfinder weapon stats here, a quiver contains 20 arrows and weighs 3 pounds. Therefore, you should be able to bring as many as you can carry, which depends on your carrying capacity and method of transportation.
Edit: A PC archer, by third level, can probably bring enough arrows with them (using a mount, a strong friend, a ship, or similar for what's to bulky for themselves) and has enough gold that it is not worth tracking how many arrows they expend in most cases.
Having carried 10 arrows per quiver with 1.5" diameter heads, and three quivers usably, doing SCA wars... and knowing that real combat arrows are under 1/2" diameter, one could reasonably shove 30+ arrows per quiver. Note that the second and third quiver was not draw-ready, but was where I could re-sling it on the field to draw from.
So, in theory, I'd say probably 120-160. But you're not going to want more than 30 in a quiver, and that quiver is several inches in diameter. It's not that heavy, but it is bulky.
I would say 20 per quiver. I did not find something specific in the rules but reading the description of Efficient Quiver it seems that a normal quiver contains 20 arrows:
This Appears to be a typical arrow container capable of holding about 20 Arrows. It has three distinct portions, each with a nondimensional space allowing it to store far more than would normally be possible. The first and smallest one can contain up to 60 objects of the same general size and shape as an arrow. [...]
I would then say that if you have a backpack you can carry no more than one quiver
As mentioned by @sebsmith and @Matteo, the rule is 20 arrows per standard quiver. When you play, you should also consider the logistics of a quiver's size, not just its weight. In real life practicality, a typical adventurer can really only carry one or two quivers at a time, making a practical limit of 20-40 arrows carried without using a mount or a henchman.
There are magical helpers as well, such as the Efficient Quiver that @Matteo mentioned (known as a Quiver of Elhonna in DnD editions), which can boost the capacity 3x or more.
Though rules vary or don't exist in the various rule-sets, our group typically will play a rule that 50% of arrows that don't hit their intended target are recoverable/reusable, which tends to lessen a little bit the problems associated with only being able to carry a finite number of arrows. Also, being able to scavenge arrows from defeated enemies helps as well.