Blackbeard is one of the Yonko of the Sea in the New World in One Piece and an incredibly mighty pirate. His rise to absolute power happened recently, following the Paramount War that ensued shortly before the two-year timeskip. At some point during the timeskip, he rose to the position of Yonko and eventually acquired even greater power by forming great alliances but, more importantly, acquiring mighty Devil Fruits.
Blackbeard's crew is known to be on a special lookout for Devil Fruits. Given that he has the ability to steal powers and even consume more than one, there are tons of Devil Fruits that he has set his eyes upon. What's more, Blackbeard's entire pirate crew has now become full of powerful ability users and they're not done with their hunt just yet. Several Devil Fruits are within their sights at the moment.
7 Mero Mero no Mi
The Mero Mero no Mi or Love Love Fruit is a Paramecia type of Devil Fruit and Boa Hancock is its current user. It grants her the ability to turn everything into stone with various close and long-ranged attacks, shaped like hearts. Most of its attacks don't require lust to function as it can even turn inanimate objects to stone, such as cannonballs. It has a specific technique, called Mero Mero Mellow attack that does work on lust.
In chapter 1059 of One Piece, Blackbeard attempted to kill Boa Hancock to obtain her fruit but failed, showing this powerful fruit is one he's been wanting for a while.
6 Hito Hito no Mi, Model: Nika
Formerly known as the Gomu Gomu no Mi for over 1000 chapters, the truth about this supposedly Paramecia fruit was revealed during Monkey D. Luffy's fight against Kaido. Its true nature being a Mythical Zoan, the Hito Hito no Mi, Model: Nika is one of the most wanted fruits because of its power and the Joy Boy title.
Blackbeard would undoubtedly want to access this overpowered Devil Fruit as its power is feared by the World Government.
5 Kage Kage no Mi
The Kage Kage no Mi is a Paramecia-type Devil Fruit whose user is the former Shichibukai that the Straw Hats defeated during Thriller Bark, Gecko Moria. This fruit grants its user the ability to control their shadows, including their own. This also includes stealing shadows by cutting them and giving them to corpses. One of its most overpowered abilities is to revive a corpse with the shadow's personality.
The last time fans saw Moria, he was on the Pirate Island, which is Blackbeard's territory. Blackbeard, having already killed his crew mate Absalom for his devil fruit, could put Moria next on the list.
4 Goro Goro no Mi
The Goro Goro no Mi is a Logia type of Devil Fruit that belongs to God Enel, an antagonist Luffy defeated in the Skypiea arc. This Devil Fruit is extremely powerful and allows the user to transform into and create electricity. According to Nico Robin, this fruit is deemed one of the few "invincible" fruits.
Enel's whereabouts were revealed in One Piece's 9th cover story where he left for outer space and met with space robots. It's currently unknown if he will resurface in the story but as of now, he is inaccessible to Blackbeard. Nonetheless, his fruit is one that Blackbeard would definitely want.
3 Uo Uo no Mi, Model: Seiryu
The Uo Uo no Mi, Model: Seiryu is a Mythical Zoan-type Devil Fruit whose current user is fallen former Yonko Kaido, the King of the Beats. He had been given this Devil Fruit by Big Mom during the God Valley incident. This devil fruit grants him the power to turn into a gigantic azure dragon, along with its other hybrid forms.
With the recent fall of Kaido, it's possible Blackbeard could take advantage of that to acquire this legendary fruit of the former Emperor of the Sea.
2 Tori Tori no Mi, Model: Phoenix
The Tori Tori no Mi, Model: Phoenix is a Mythical Zoan-type Devil Fruit consumed by the former right hand of the late Whitebeard, Marco the Phoenix. The Devil Fruit allows its user to turn into a blue flame Phoenix with incredible healing and restorative abilities, among others.
Blackbeard would be keen on collecting more Zoan-type Devil Fruits and seeing his history with Marco during the Payback war, this Devil Fruit could be high on his list.
1 Mera Mera no Mi
The Mera Mera no mi, or Flare Flare Fruit, is a Logia that belonged to Portgas D. Ace before his death. Now having been inherited by Sabo, his brother, this fruit grants its user the ability to create and transform into fire.
While Blackbeard has had the opportunity to kill Ace after their fight and steal his Devil Fruit, it benefited him more to hand him over to the Marines at the time. Thanks to that, he became one of the Shichibukai. In Dressrosa, however, Teach sent Jesus Burgess to win the fruit for him in the battle in Corrida Coliseum. While Burgess failed, it gave fans a confirmation that this fruit was one the Yonko was heavily interested in.