The Hybrid

The Hybrid was a Gallifreyan legend telling of a creature thought to be crossbred from two warrior races - the Time Lords and the Daleks - according to prophecies from the Matrix. Rassilon was searching for information about the Hybrid as according to the legend it would stand over the ruins of Gallifrey before proceeding to unravel the Web of Time. (TV: Hell Bent)

Possible Interpretations of the Hybrid
In all possible theories of its identity, the Hybrid refers to itself as "Me." This leads to several possible interpretations as to who it is based on the theories mentioned above. The three most prominent candidates for being the Hybrid are the Doctor, Ashildr, and the Doctor and Clara put together.

Ashildr
When Ashildr died after helping him and her village fend off the Mire, the Doctor resurrected her using a Mire medical kit. The result was Ashildr becoming immortal and losing the "ability" to die. (TV: The Girl Who Died) The next time the Doctor encountered her, Ashildr was calling herself "me", after living through centuries of losing everyone she ever loved. From lovers to children, she continually lost people she cared about until she became callous and uncaring towards others and bitter towards the Doctor for condemning her to such a lonely existence. She told the Doctor that the name she called herself now was "Me", since there was no one else to call her "Ashildr", and that because she would be the only constant in her life. (TV: The Woman Who Lived) When the Doctor met her again, Ashildr was standing in the ruins of the Cloisters, a chamber underneath Gallifrey - fulfilling that part of the legend. The Doctor likened her to a hybrid, explaining that she was a result of two warrior races - human and Mire - although the apter description was the former with a bit of the latter. (TV: Hell Bent)

The Doctor
While the death of his companion Clara Oswald was still impending, the Doctor threatened Ashildr with exposure to the world if she did not do anything to save her. When Ashildr pointed out that the Doctor would never do such a thing, the Doctor's chilling reply was "The Doctor is no longer here, you are stuck with me now!" It's possible that the moment Clara died was the moment the Doctor became the Hybrid. (TV: Face the Raven) As stated above, when the Doctor emerged from his confession dial onto Gallifrey, he explained that the Hybrid couldn't be half-Time Lord-half-Dalek as the Daleks would never allow a half-Dalek being to exist given their ideas on racial purity. In keeping with the theory, the Doctor - donning his sonic sunglasses - disclosed that the Hybrid was "Me." (TV: Heaven Sent) The Doctor did indeed end up standing over the ruins of Gallifrey when he met Ashildr again at the end of time. (TV: Hell Bent) In addition, the Doctor's four-and-a-half billion years within the confession dial involved him -- having figured out its secrets and near death at the hands of the Veil -- using his dying body as an energy source to reactivate the teleporter that originally brought him there and cause his younger self to materialize once more, starting a cycle repeated billions of times until he made his way through the Azbantium wall that stood between him and freedom. (TV: Heaven Sent) He undertook this ordeal only to avoid making a final confession about the Hybrid and thus have a bargaining chip to use with the Time Lords in his plot to save Clara and end his anguish -- perhaps reflecting a key detail of the prophecy that the Hybrid would "destroy a billion billion hearts" in the process of healing an unspecified wound to its own. In the end, succumbing to a memory wipe, the Doctor told Clara he accepted his fate, noting "I became the Hybrid." (TV: Hell Bent)

The Doctor and Another Person
When the Doctor met Ashildr again at the end of the universe in the ruins of the Cloisters, she told him her theory that the Hybrid was not one person but two, the second being either Clara or the Master. The Doctor dismissed this theory along with another theory she had where the Doctor ran away from Gallifrey because he was afraid of himself. (TV: Hell Bent)