Genie only officially granted the wish to be made free

Of the other two wishes, one was in progress but never fulfilled and the other was not official since it was never verbalized.

Wish 1 - Genie never makes Aladdin a prince

Aladdin wishes for Genie to make him a prince. Genie then grants him a menagerie and all sorts of other fancy things to make him look like a prince, but he is not officially a prince as is pretty well established throughout the rest of the movie.

The only way for Aladdin to become a prince is to marry a princess. So Genie is still in the process of granting Aladdin's first wish the entire movie.

Genie tries to convince Aladdin to tell Jasmine the truth about not being a prince in order to woo her

Genie: All right, Sparky, here's the deal. If you wanna court the little lady, you gotta be a straight shooter. Do you got it?
Genie: Tell her the truth!
Aladdin: No way. If Jasmine found out I was really some crummy street rat, she'd laugh at me.

Jafar exposes Aladdin as a non-prince

Jafar: ? Prince Ali, yes, it is he. But not as you know him! Read my lips and coms to grips with the reality. Yes, meet a blast from your past, whose lies were too good to last! Say hello to your precious Ali! ?
Iago: Or should we say Aladdin?
Jasmine: Ali?
Aladdin: Jasmine, I tried to tell you. I'm just...
Jafar: ? So Ali turns out to be merely Aladdin! Just a con. Need I go on? Take it from me.

The Sultan has to make a new law so that Aladdin and Jasmine can get married

Genie: No matter what anybody says, you'll always be a prince to me.
Sultan: That's right. You've certainly proven your worth as far as I'm concerned. It's that law that's the problem.
Jasmine: Father?
Sultan: Well, am I Sultan or am I Sultan? From this day forth, the princess shall marry whomever she deems worthy.

At this point Genie has already been set free so it could safely be said that he has not granted Aladdin's wish of being made a prince before being set free. A change in the law actually has to be made so that Jasmine and Aladdin can get married. Its also worth noting that they do not get married until the third movie Aladdin and the King of Thieves

Wish 2 - Aladdin doesn't officially wish to be saved from drowning

Genie can't grant an official wish unless Aladdin says the words, "I wish for [blank]", but this doesn't limit Genie's ability to intervene. There are two examples in the movie of Genie performing acts on Aladdin's behalf which Aladdin didn't verbally wish for.

Unofficial wish #1

First, Aladdin tricks Genie into getting them out of the Cave of Wonders, which Genie accepts was not an official wish due to this logic.

Aladdin: Ah, no. I never actually wished to get out of the cave. You did that on your own.
Genie: Well, I feel sheepish. All right, you baaad boy, but no more freebies.
Aladdin: Fair deal.

Unoffical wish #2

When Aladdin is drowning, Genie saves him but since Aladdin is unconscious he is unable to make an actual wish. Genie even tells him that he can't save him unless he says the words. But this would appear to be untrue as he immediately goes ahead and saves him anyway after Aladdin appears to nod.

Genie: Never fails. Get in the bath and there's a rub at the lamp. Hello? Al? [gasping] Al! Kid, snap out of it. You can't cheat on this one. I can't help unless you make a wish. You have to say, "Genie, I want you to save my life." Got it? Okay! Come on, Aladdin! I'll take that as a yes.

Counter arguments

It has been argued that this second wish was official because of the original agreement of "no more freebies", however it is already established that Genie does have the ability to intervene without there being an actual wish made. I don't see any evidence in the film or otherwise that the "no more freebies" agreement was binding in any way that would eliminate Genie's ability to perform an action without a wish being made.

In addition, I would argue that Genie was already obligated to save Aladdin from drowning due to Aladdin's wish to be made a prince which had yet to be fulfilled.

Conclusion

So, Genie never grants Aladdin's first wish to made a prince. Aladdin only becomes a prince after he marries Jasmine, which he doesn't do until the third movie and is only able to do because the Sultan changes the law.

The wish to save Aladdin from drowning was non-binding because:

• Genie was obligated to save him because he had not yet fulfilled the first wish to make Aladdin a prince
• While Genie and Aladdin had a verbal agreement of "no more freebies", it is established that for the wish to be official, Aladdin must say the words, which he doesn't do.
• It is also established that Genie does have the ability to intervene without a wish being official

Aladdin already was a prince, even before he made the wish.

From the third movie we know that Aladdin's dad is the king of thieves, which sort of makes him a prince. Of course you can still argue wether the title was legit or not, but what defines a king anyway? If it's the people under his reign, he had those. (39 thieves I think)

Aladdin already was a prince, so basically his wish would still have been fulfilled even if Genie had done nothing. But that would have been a quick end to the movie and a major spoiler I guess.

So the answer is no, Genie does not own Aladdin a wish.

Genie does grant the wish to make Aladdin a prince!

The easiest counter argument is that without Jafar's involvement Aladdin would have stayed a prince. Jafar's magic counteracts Genie's when he "exposed" Aladdin.

Aladdin was made a prince, the "real truth" he needed to tell Jasmine was that he made a prince by magic and that he was not born royal. Remember Jasmine was mad at Ali because he felt he had a "right" to marry her. Telling her the truth at that point reveals that the whole prince thing is just a way to even have a chance at seeing her again.

Jasmine ends up realizing Ali and Aladdin are the same person, but Ali ends up covering this with another lie. If Jasmine believes this lie she still believes he is a prince. Without Jafar's interference, if Aladin can reestablish the connection he made with Jasmine in the market there is no need for the Sultan to change the laws as the secret would not be made public (or at least to public to the Sultan).

The conclusion would be that Aladdin and Jasmine get married, Genie fulfilled the wish.

Aladdin got exactly what he was looking for with all three wishes.

Let's start with the first wish, looking at it in its full context from the Genie's perspective.

You're a genie. You've just met your new master, a kid named Aladdin. Dirt-poor, judging from his clothes -- a real underdog. Sneaky and clever enough that he's tricked you into helping him escape for free, but you can't entirely blame him for you not paying attention, and he seems to have gracefully accepted that he won't be able to do that again. Good-natured enough that he was willing to ask what you would wish for, which nobody ever has, and offered to give it to you with his last wish. So on balance, you like this guy, enough to do right by him. So what does he want?

Aladdin: Well, there's this girl--

Genie: Eehhh! Wrong! I can't make anybody fall in love, remember?

Aladdin: Oh, but Genie. She's smart and fun and...

Genie: Pretty?

Aladdin: Beautiful! She's got these eyes that just... and this hair, wow... and her smile! Ahh... but, she's the princess. To even have a chance, I'd have to be... hey. Can you make me a prince?

He wants to be able to court this princess, and wants to be a prince to that end. Meaning for the purposes of this wish, a prince is someone who is recognized as a prince -- by the Sultan, by Agrabah, by Jasmine.

It's a facade, but a facade is exactly what Aladdin's looking for -- and of course, facades and hidden natures are all over the place in this movie. A piece of brass junk hiding one of the most powerful beings on Earth; a thieving street rat who proves to be a diamond in the rough; a respectable royal adviser hiding dark ambitions; a birdbrain of a parrot who's secretly as intelligent and devious as his master (and an expert in disguising his voice); a woman in rags hiding a sheltered princess hiding a strong will and quick wits. In the movie's very first scene, the merchant (who may well secretly be the Genie) says, "it is not what is outside, but what is inside that counts" -- and Jafar later confirms (while in disguise, naturally), "Things aren't always what they seem".

Aladdin doesn't get a kingdom to go with the parade and clothes and elephant because he's not looking for a kingdom. Could the Genie have taken that route? Sure he could've. In fact, at the end of the movie, he was probably planning to:

Aladdin: Genie, I wish for your freedom.

Genie: One bona fide prince pedigree, coming up! I -- what?

By this point, of course, the Sultan knows who Aladdin really is, so it's reasonable for the Genie to assume a little more work would be necessary to satisfy the Sultan and the law; some documents certifying lineage, maybe a little history-rewriting.

Still, that's not what Aladdin wanted when he made his wish. He wanted, and got, a facade. True, later on he becomes uncomfortable with that fact -- the knowledge that everything people love him for came out of a lamp -- and grows to fear Jasmine finding out about his actual life history. (Also the Sultan, but I get the sense Aladdin's more concerned about Jasmine's approval than the law's.) Even if he did have a kingdom somewhere, it arguably wouldn't matter to his arc -- he'd still know he wished his way into it, and be tormented by hiding that fact from Jasmine and Agrabah.

Now how about the second wish? Clearly, the Genie fudges things a little. Maaaaybe Aladdin was still just barely conscious enough to move his head. Maybe he still had the presence of mind to nod in response to the Genie's prompt. More likely, he had already passed out by that point.

Two things, though. One, Aladdin had clearly rubbed the lamp with intent to wish himself to safety. Two, neither he nor the Genie dispute that it was a legitimate wish. They both let the "official version" stand. The Genie pretty much had to interpret it as a nod -- now that he knows enough to not let Aladdin cheat, it seems he can't let Aladdin cheat -- and Al's not nearly enough of an asshole to try to rules-lawyer it.

The point of all this is, the system isn't a machine. The two of them are bound by the rules, true (though the Genie isn't blocked in the cave, when he legitimately fails to realize no wish has actually been made). But the Genie has discretion in how he interprets wishes. And if he and his master have a mutual understanding afterward that a wish was made and granted, it works out, even if it's essentially a fiction like the second wish.

The Genie even shows some of this discretion with Jafar's first wish. If he can give Aladdin a "bona fide prince pedigree", presumably he can change things around so that everyone remembers Jafar having been the Sultan. Instead, he does the bare minimum -- in this case, another facade (useless under the circumstances), dressing Jafar in the Sultan's clothes. All anyone has to do is refuse to respect his authority, as Jasmine demonstrates. Regrettably, "the most powerful sorcerer in the world" and "an all-powerful genie" have less wiggle room.

(In The Return of Jafar, the fact that genies have wiggle room is more clearly displayed... on the other hand, Jafar suggests that unlike our own Genie, he can give things that weren't at all wished for. Though Abis Mal suspects that those treasures will disappear after he frees Jafar, and he may be right -- "Friend Like Me" clearly shows that even our Genie can conjure phantasmal treasure for demonstration purposes.)

The third wish, of course, is unambiguous and undisputed.