I've been researching the Walking Ways and I came across this one. The theory goes like this:

When multiple souls combine together, they become a divine being. Each soul has power. If they stack on top of each other, that power increases by the number of souls in the mix. The effect is even greater when the souls are more powerful. There are many examples of this, and it is common among the divine beings in the Aurbis.

Notice that the Divines exist within an Oversoul. The best example of this is Time. Aka. Akatosh. Alkosh. Auriel. Bormahu. On and on the list goes.

The same is true about the ALMSIVI. They were three beings but one soul. The same with Talos. The Numidium, likewise, is an incredibly powerful entity that was made from the merging of thousands upon thousands of Dwemer souls.

Now, we get to the interesting implications for Dragons. This could serve as an explanation for Alduin. His power is, as one poster put it, "The Highlander's Prize". Alduin was likely once a normal dragon, but he figured out the power that can be gained by consuming souls. Hence his hunger for them in Sovngarde. He eats souls to gain power until he is far stronger than any other dragon in existence, strong enough to eat the entire world.

What other evidence is there for this? See the "Seven Fights of The Aldudagga".

I bolded that part on purpose. What are the Six Walking Ways? Methods to reach Heaven by violence. Alduin was a normal dragon who became a god, not a god that became a dragon.

This also has huge and interesting implications for the Dragonborn. If Alduin became a god by eating all the souls of mortals, what happens when a Dragonborn absorbs all the-considerably more powerful-souls of dragons?

http://www.youtube.com/watch?v=m8k8kBg_Hig

He becomes Alduin?

I sure hope so, since that's my current head-canon.

Miraak was utterly convinced that if he could absorb the PC's soul he'd have enough power to finally break out of Apocrypha, which he thinks Mora could do nothing about. Certainly seems that the number of souls a dragon absorbs strengthens them on some level aside from expanding their own knowledge.

The PC runs the risks of becoming...something, but we flat out don't know what's really involved in that aside from divine hijinxs. Add to the fact that the PC's fate is rarely touched upon aside from a vague reference and rumor, their becoming Alduin, or anything else for that matter, is regulated into the realm of head-canon.

EDIT:

Not entirely sure how true this statement is, at least during the time of the Dragon War. Felldir flat out says that Alduin cannot be harmed/slain by normal means, since he is not like "lesser dragons." There seemed to be something different about him to begin with, and his innate name most likely confers the fact that he is destined to devour everything regardless.

Is there anything to the mortal LDB eating immortal souls, while the immortal Alduin eats mortal souls? At the very least I like the duality between the two.

Dragons can take each others' souls, too. It's just not something they tend to do because, well, they can't reproduce to replenish their numbers. They fight for dominance, not to kill. Depending on how you play, that can also apply to the LDB (much to Esbern's and Delphine's chagrin).

I thought Alduin was the Nordic aspect of the Aka Oversoul.

I also like the idea that the LDB becomes Alduin, at least in a metaphysical sense. After all, a World Eater must exist to trigger the next Kalpa, and the cycle is too deeply ingrained with the nature of the Aurubis to simply stop.

Vehk renamed himself Vivec when he became a god and Tiber Septim became Talos upon his ascension. Is there any reason Alduin couldn't have renamed himself once he became the World Eater? I don't think Alduin has any innate divinity, or at least no more than any other dragon has. He acquired his divinity by means of soul absorption, until his hunger got to the point he defined the ending of time. Thus he became an aspect of Aka.

You can't harm him without Dragonrend. And even with it, you couldn't kill him (the first time he was able to run away to Sovngarde, second time it took some kind of intervention). Alduin is also the only one that seems to know Slen Tiid Vo, to bring back slain dragons. He's definitely more than a typical dragon.

But was Alduin a dragon that innately had those abilities, or did he gain them overtime from his devouring and feasting?

According to in-game dialog, Alduin's problem was that he wasn't devouring, and instead was trying to rule.

As blai said, that doesn't make it an innate trait. He could just as easily have gotten that way from gorging himself on souls.

His problem is directly relevant to the question, though. The question is if he got those traits from devouring, and his problem is that he wasn't devouring like he was supposed to, which would suggest the answer to be no. He wasn't devouring, so that's unlikely to be the reason why he's "more immortal" than a typical dragon.