Throughout Avatar: The Last Airbender and The Legend of Korra, there were various abilities that the Avatar never learned. Such special skills are seldom dived into in terms of how they function in the world, and it remains a question whether or not these are inherent abilities benders are born with or if they are teachable talents. While Aang picked up some talents, like Toph's seismic sense or Zuko's ability to redirect lightning, he left it to his successor Korra to add metalbending to the Avatar's repertoire.
What if a future Avatar were more comprehensive in adding such rare abilities to their spiritual line's pool of knowledge? In many cases, the Avatar's reason for neglecting to learn such skills seemed plot-reliant -- obviously, some supporting characters become redundant if the Avatar can replicate all their abilities -- but that doesn't mean a future Avatar won't take a more logical approach to their duties. The Avatar is already the most powerful bender in the world, but with these abilities, they would become the most powerful bender in history.
Spiritual Projection
As far as special sub-bending skills go, spiritual projection is hard to think of as one of the most valuable. Most prominently featured by Jinora in Korra, the airbender's spiritual attunement allows her to meditate and transmit her spirit to remote locations. She used the ability ably in tracking down Kai by using her personal attachment to him, but meditating in the middle of a fight isn't exactly the smartest move. However, it's the skill's utility outside of a fight that would make it so valuable to the Avatar.
Despite all their other powers, the Avatar is most often limited by the fact that they're still a single mortal individual in a vast world. They can only be in one place at a time, and if they cannot find the places they need to go, no amount of bending in the world can help them. However, as Aang demonstrated with the Banyan Grove Tree in "The Swamp," the ability to spiritually attune to a target is an invaluable skill. Aang is probably spiritual enough to learn the skill, but in a time before Jinora, he likely didn't even know it was an option. A future Avatar would do well to gain the ability -- if they did, no foe could hide from their wrath.
Lightningbending
The first Avatar to come across lightningbending was Kyoshi when she faced the Firebender Xu Ping An in a duel to the death. The skill immediately proved useful when Xu Ping An fired it off before Kyoshi could take any action to respond, and it proved equally effective in following eras where the Fire Nation's royal family used it to devastating effect. Azula even managed to kill Aang with the technique, nearly ending the Avatar Cycle altogether before Katara brought the last airbender back to life.
This is perhaps the most puzzling skill for the Avatar to never learn even by the Era of Korra. In that time, it's shown to be such a common practice that Republic City fuels itself on blue-collar firebenders who generate lightning for the city's power supply, proving that it's a teachable talent that many can learn. Considering its devastating speed and power, it would be a valuable addition to the Avatar's arsenal, and given that Aang already learned lightning redirection, it's a natural extension of what the Avatar already knows.
Lavabending
Unlike any other skill, lavabending occupies an interesting place in the Avatar's skillset because it's technically something they're already capable of. Avatar Szeto demonstrated the ability to erupt multiple volcanoes at once, Roku carved through islands on multiple occasions with the skill and Kyoshi even managed to create Kyoshi Island after summoning a wall of lava from the earth. However, the trick here is that all these showings required the Avatar State to pull off, and for that reason is believed to require the use of both earthbending and firebending to perform successfully.
Nonetheless, this is not a necessary requirement to lavabending, as both Ghazan and Bolin prove in Korra. The earthbenders both phase-shift solid stone into lava without any Avatar State required. Learning the ability without the need of accessing the Avatar State would mean an Avatar could use it far more easily. The question is whether or not any bender can learn the skill, as Bolin's best efforts still barred him from the ability to metalbend. They may be mutually exclusive abilities given that Ghazan is also not shown metalbending, although such limitations may not apply to the Avatar.
Combustionbending
Another ability that could well be genetic given it's so rarity -- only two benders ever demonstrate it in the entire franchise -- is combustionbending. Although their connection is not known, the mercenary Combustion Man of the original series and the Red Lotus member P'Li from the sequel series could well have been related. At the least, they are known to share the common ability to project their firebending through the third eye tattooed on their foreheads, creating an explosive force more destructive than even the highest level of bending attacks.
Armed with the ability, Combustion Man proved not only a match for Aang but him and his friends all at once. Considering Toph and Katara were arguably the greatest earthbending and waterbending masters respectively of their day, it speaks to the skill's unbelievable power that the mercenary could fight them and the Avatar off at once. The only way to keep the Avatar line safe from the skill in the future is for the Avatar to learn it for themselves. With its awesome power, there's no telling the level of destruction they could wreak.
Bloodbending
As powerful as all the previous abilities may be, none of them hold a candle to what bloodbending could do in combination with the Avatar's raw strength and skill. By utilizing the water within a target's blood, waterbenders are capable of bloodbending their targets so they can manipulate their bodies like puppets. Hama first demonstrated the ability against Katara, who immediately learned the ability and dedicated much of the rest of her life to banning it throughout the world.
However, as Tarrlok and Amon both demonstrate in the sequel series, even the Avatar is helpless when their own limbs are turned against them. A flashback to the bloodbenders' father Yakone paralyzing an entire courtroom proves that even the strongest bloodbenders are useless when the Avatar State activates and makes them immune to its effects; nonetheless, for the ability for an Avatar to use the ability offensively, and to do so with the continent-destroying power of the Avatar, would mean that any user dark enough to turn to the ability would inarguably be the most powerful human in the Avatar universe.
