Why did the Free Cities put up with the Ironborn

[blood of the dragoon]

I imagine that it's somewhat like the free cities approach to the dothraki. Cheaper to just pay them off.

Though i doubt a naval superpower like braavos would put up with them.

[Nucky Thompson]

Still takes time. The Greyjoy rebellion was 289, a game of thrones starts in 298, only nine years later, and a clash of kings is set during 299. They must've been working pretty hard to restore their old strength in those years.

Well, it's what they do. Without a fleet threatening everyone else, they don't belong in the story. Moreover, who is to say that they didn't send half of their ships off westward when they saw that defeat was imminent and just tell Stannis and Robert that there was a storm, that the captains were cowards, or whatever. It is clear that the Baratheon regime didn't conduct a thorough investigation of the Greyjoy naval power after their submission.

Tax profits from a few islands that are barely able to feed themselves without resorting to plundering?

It seems unlikely to me that the King would let them off the hook so easily without collecting any taxes - what does his power over them amount to, then? Furthermore, the collection of taxes from the Crown might be the very reason for their inability to feed themselves.

[Rahgh Neqaj]

Well, it's what they do. Without a fleet threatening everyone else, they don't belong in the story. Moreover, who is to say that they didn't send half of their ships off westward when they saw that defeat was imminent and just tell Stannis and Robert that there was a storm, that the captains were cowards, or whatever. It is clear that the Baratheon regime didn't conduct a thorough investigation of the Greyjoy naval power after their submission.

They were doing quite well until the moment they engaged Stannis' fleet. Back in the times, a fleet battle was fighting at close quarters - the best you had in terms of ranged weapons was fire arrows and catapults which both aren't very far ranging. So any ship wishing to make off would have to disentangle itself and then flee faster than any pursuer can follow. Now Stannis and his men would have noticed if there had been reasonable numbers of ships making off during battle and they would have pursued. Also keep in mind that Euron is the only one of all Planetos sailors that we've seen so far who doesn't mind to sail far from the shore and he's rumoured to have sacrificed to the big & bad stormgod in order to do so. Finally, strategic retreats simply don't fit the Old Way. Anyone doing a retreat would've been considered a coward and would have faced serious leadership challenges.

I fully agree that they were busy rebuilding their fleet and preparing youngsters to go reaving but that takes time. And yes, they probably helped themselves to some wood from the north-western shore but even that takes time as the wood has to be cut, dried, possibly hardened and transported. (I'm no expert on ancient ship building but I vaguely remember that wood for ships has to be hardened or something over fire and that it had to rest for some time before being usable unless you want it to twist apart and soak water as soon as you put it in the sea.) Also, the western shore of the North isn't quite as unpopulated as one might think. There's the Glovers of Deepwood Motte and the houses sworn to them, the Tallharts, Ryswells and Dustins south of the Wolfswood and the Flints of Flint's Finger on Cape Kraken. Certainly these would have noticed and raised alarms had the Ironborn started large scale wood cutting.

It seems unlikely to me that the King would let them off the hook so easily without collecting any taxes - what does his power over them amount to, then? Furthermore, the collection of taxes from the Crown might be the very reason for their inability to feed themselves.

Any good king would want to raise taxes from the ironborn. Problem is that there's not much to raise on those islands unless they go reaving for which they need the fleet and which they've been forbidden to do.

A final thought: They slapped good Tywin Lannister in the face when they burned his fleet inside the harbour of Lannisport. Tywin must've been bloody annoyed by it and I'm pretty sure he'd have whipped their asses a lot harder than Robert, Stannis and Ned did. (A lot more Castamere like.) Tywin's daughter happens to Robert's Queen - she probably remained in KL during Robert's campaign so she wasn't able to influence Robert when it came to punishing old Balon but I very much doubt she was happy to see them get of that easily. When it comes to punishing, she shares Tywin's views and she certainly identifies primarily as Lannister, not as Robert's wife. Now I know a king who's trying his best not to annoy his queen too much except when drinking and whoring. So would that king, out of the goodness of his heart and due to his keen interest in counting coppers, sell wood to the Ironborn or help them in any other way possible?

[Bright Blue Eyes]

Still takes time. The Greyjoy rebellion was 289, a game of thrones starts in 298, only nine years later, and a clash of kings is set during 299. They must've been working pretty hard to restore their old strength in those years.

At the technological level, the average life span of a ship was ten to fifteen years. Between Balon's Rebellion and the War of the Five Kings, the Ironborn would have replaced almost all their ships anyway, no hard work necessary to restore their strength, just a tad above the average.

[Ser Lepus]

It seems that, with a few exceptions, everybody is raiding everybody in the ASOIAF world. There are myroshi and lysene pirates in the Stepstones, Basilisk Island's Corsairs, Ironborn, Slaver's Bay's slaver galleys, volantene slavers who aren't above taking passengers and selling them...etc.

Some people, like the lysene and the myroshi are pirates themselves, and others, like the tyroshi, are too busy trying to stop the lysene and myroshi pirates and can't find time to take care of a few ironborn attacks.

Truth to be told, what I find most surprising is that there is such a great amount of trade despite the many pirates. You would think that nobody would want to invest money in building ships and buying cargo when the chances of getting seized are so high.