Fairies count as "insects" in the game code, and insects with fixed spawn locations (like for example the Energetic Rhino Beetles in Kakariko) have a 1% chance to respawn for every minute of IRL active playtime ("Random Revival"). This means that you can collect every such insect once every 3-4 hours on average, and the same applies to Fairies - in theory.

Nintendo apparently doesn't want players to just stock up on a hundred Fairies in addition to Mipha's Grace, "Hearty" meals and the ability to heal whenever you want without a penalty. But instead of lowering the max count you can carry (999 in theory) they implemented a different mechanic: a spawn limit.

If the player already has 3 or more Fairies in their inventory, the "wild" Fairies around the fountains won't show up even if they're already respawned. Think of them as being there, but invisible. They should exist, but you can neither see nor catch them right now.

The game only checks for the Fairies in your current inventory tho, and there are some neat tricks to use this to your advantage.

Up to 11 Fairies (Difficulty: Easy)

Find a Fairy Fountain that hasn't been looted recently, and stay close enough that you can see the Fountain but not so close that you would immediatly scare away freshly spawned Fairies. Open your menu and hold up to 5 Fairies in your arms, then close the menu while sill holding them. If you had less than 3 in your inventory for a couple of seconds, the "invisible" Fairies around the fountain will pop into existence, and you can then put the ones you're currently holding back into your inventory and go catch the freshly spawned ones.

This only works if you can get the "inventory Fairy count" below 3 while holding them in your arms, so it might even be benefcial to release excess Fairies, as counter-intuitive as it may sound. If you, for example, have 8 Fairies at the moment and you hold 5, the inventory count only drops down to 3 and no new ones will spawn. If you release one of them and then hold 5 in your arms, the inventory count will be below 3, the "invisible" Fairies will appear, and if you catch them you'll end up with 11 total (2 inventory + 5 held Fairies + 4 new ones) so by releasing one you gained three more than you formerly had.

Up to 16 Fairies (Difficulty: Medium)

A bit more risky but still possible, is that you release some excess Fairies, hold 5 more in your arms, immediatly open the menu again once the invisible fairies pop into existence, and then quickly catch the released ones again. Triple Stealth and careful movements help with this, but it is still risky as the released Fairies are prone to zooming off as soon as they can.

If you start out with 12 Fairies, "release" 5 of them, hold 5 more, and then catch the 4 newly spawned ones in addition to the ten that were "on hold", you'll end up with 16 total as the max.

Up to 21 Fairies (Difficulty: Expert)

This requires precise timing, and it is advised to save the game before attempting it, so you can reload if you mess up - otherwise you might end up losing a bunch of Fairies in the process.

Stay a bit farther away from the fountain but within sight range, and make sure that there is empty space (and no cliff!) behind you and that you have one of the bomb runes equipped... and enough health to survive 1-2 blasts. Place down a (cube) bomb, hold up to 5 Fairies in your arms. Release them, and immediatly blow up the bomb.

This will hurl Link away from the Fairies, which will continue floating around where he dropped them, oblivious to the explosion. There is a grace period of approximately one second in which the Fairies can not be harmed or even startled, and since Link is too far away after the explosion, they won't flee.

You can then do this a second time to "release" up to 10 Fairies at once, and then hold 5 more in your arms - but do not release those 5 as well, because if Link drops more than 10 "items" at once, the oldest ones will vanish and this would delete the first 5 Fairies from existence.

If done right, 4 more Fairies will spawn, 10 "released" ones will be floating around you and can be caught again, 5 are in your arms, and 2 are in your inventory, for a total of 21 Fairies.

Please note that, whatever method you end up using, the Fairies will still need to respawn first in order to farm them again. You can not just immediatly release all inventory Fairies again and expect new ones to appear a second time. After you collected the ones you made appear, you have to wait around 3-4 hours for them to respawn into that "present but invisible" state or find a different Fountain that hasn't been looted recently.

Additional fun fact: While you have less than 3 Fairies in your inventory, there is a small chance (IIRC 0.2%) for any random insect on the map to be swapped with a fairy, meaning that you can find them while cutting grass and lifting rocks, but also chilling in darner swarms and fluttering around where butterflies or Restless Crickets may appear. This simply doesn't happen while you have 3+ Fairies already, and since players usually stock up on Fairies as soon as they can and maintain a decent supply, it almost never happens to them ... unless you're doing a Fairy-free challenge run, then those pesky pink puffballs won't leave you alone.