The mission "Devil's Advocate." Getting to take the helm of the infamous Queen Anne's Revenge and all forty of her guns.
Topped when you get to steal a Portuguese Man of War and get to put all its awesome firepower to use. The ship ultimately goes on to become Black Bart's Royal Fortune. For added awesome value, you get to defeat that ship in the final sequence of the game.
The fact that the Animus doesn't desynch if Edward dives off of the Jackdaw, swims towards an enemy ship, boards and captures it single-handedly — thereby foregoing the need for a naval combat against that ship — means that it's entirely possible that he actually did this at least once!
Really, the sheer amount of freedom you have in this game. Going purely by the Animus's mechanics, it elevates Edward's reputation far above what's stated. In his youth, he could dive deep in a diving bell, rob chests from under the shadows of sharks and live to tell the tale. And he's harpooned Great White Sharks and whales. And he's defeated several legendary ships, amongst many other lesser ones.
Edward Kenway is a self-trained pirate able to completely bluff Templars as he uses hidden blades for the first time.
Props to Benjamin Hornigold who, in Kingston, has one glimpse of the Jackdaw and immediately knows that Edward is actively spying on him and the other Templars, then immediately calls him out.
Taking down a Legendary ship. Even with a fully upgraded Jackdaw, you can't just trade broadside volleys with them; you have to outmaneuver them instead.
One such vessel is La Dama Negra, a huge Man O'War with a black tough armour like a rhino hide. Its weak spots are on the rear and its broadsides are deadly and its mortars are not only longer in range but also don't leave any marking on the waters as a warning sign. Finally turning this ship into a wreck and watching it sinking makes you join in the cheers of your digital crew.
There is a growing consensus that El Impoluto is the most badass of all the Legendary Ships and the most difficult to take down. While it lacks the impenetrable armor and mortars of La Dama Negra and its cannons aren't quite as powerful, it makes up for it by being stupidly fast and insanely maneuverable, effectively negating the one advantage the Jackdaw generally has over all other ships. It also seems to be crewed entirely by rum-crazed lunatics who have made it their mission in life to park their ship on top of yours, hence their fixation with chasing you down and ramming you into oblivion. With all the other Legendary Ships you have at least a decent chance of taking them down on the first try, especially if your ship is fully upgraded. But El Impoluto? No. El Impoluto laughs at your feeble mortars and cannons and fire barrels. You are all but guaranteed to die the first time you foolishly decide to challenge El Impoluto, and probably many more times after that. Which makes the day when you finally manage to send that rotten bastard down to Davy Jones' Locker all the sweeter.
The legendary ships are so awesome, not only the Spanish ships but also the H.M.S. Prince and the Dual Boss Royal Sovereign and H.M.S. Fearless that you can help but feel a bit of regret and Worthy Opponent sentiment when you sink them down, not unlike taking down bosses in Shadow of the Colossus. You wished you had the boarding option to make them part of your fleet at least, but no these proud ships are piloted by true servants of their respective empires and would never sail under the black flag, preferring death instead.
"In honest service there are thin commons, low wages, and hard labour. Yet as gentlemen of fortune we enjoy plenty and satisfaction, pleasure and ease, liberty and power... so what man with a sensible mind would choose the former life, when the only hasard we pirates run is a sour look from those without strength or splendour! Now, I have been among you six weeks, and in that time I have adopted your outlook as my own, and with so fierce a conviction that it may frighten you to see your passions reflected from me in so stark a light. But... if it's a captain you see in me now, aye then... I'll be your bloody captain! For I have dipped my hands in muddied waters, and withdrawing them find 'tis better to be a commander than a common man!"
James Kidd or Mary Read is awesomeness itself. When was the last time a woman/love interest got to be not only The Mentor to the hero but also to another woman, Anne Bonny, who she empowers from a waitress to a fellow Back-to-Back Badasses and who serves as her Number Two while she Captains Jack's operation more or less. And while she's pregnant. Alas Too Cool to Live.
Adewale is pretty awesome. He's the Only Sane Man, down and earthy, and a good friend. But he won't be Edward's enabler and tells him straight up, that he'd rather by an Assassin than sail under a selfish asshole like him. It's great to see a character who blossoms from The Lancer into a Hero of Another Story in the same game.
Retroactively, Woodes Roger's assassination is this, as at the end of the game it's revealed he survived.This makes him the second person to have survived a direct hidden blade assassination, in addition to Shakulu. And more than that, Shakulu had heavy armor; Rogers had none of that.
The whole randomness of the game world can cause some truly awesome fights, like attacking an island fort during a storm while a water spout circles it, messing up with your aim and forcing you to a distance where it's even harder to make your shots hit... unless you have the luck, timing, and/or bravado to get in close and attack from the opposite side as the deadly vortex. If you're lucky, the spout might attend to any enemy ships trying to interfere with your assault.
It's very much possible to defeat El Tiburon with nothing more than a pistol (or four). This in turn crosses into Fridge Brilliance: El Tiburon is a heavily armored brute who fights more like someone from Ezio's time (though he also has a pistol of his own), but that means nothing when Edward can just whittle him down him with bullets without ever crossing blades. In a way, it shows just how much the old forms of combat have given way to newer ones.