Crookshanks is actually Lily Potter's cat.
- In The Deathly Hallows, when the trio are in Grimmauld Place, Harry reads a letter hand-written by Lily. In the letter, she mentions that they had a cat and that Harry almost ran over it with his toy broomstick. Harry even wonders if it survived the attack by Voldemort and if it was alive.
- In Prisoner of Azkaban, the owner at Magical Menagerie from where Hermione buys Crookshanks tell her that she had had Crookshanks for "quite some time" and that nobody ever wanted him. This could correspond to our earlier timeline where the cat disappeared after Voldemort's attack on the Potters' residence.
- Soon after Crookshanks is bought by Hermione, he quickly recognises Scabbers and the shaggy dog as Peter and Sirius respectively. The fan theory goes that Crookshanks recognized them both because they would have frequently visited Lily and James when he was with them.
- There is also a popular theory (no doubt started by Tumblr) which states that the Boa Constrictor that Harry helped escape in The Sorcerer's Stone is actually Nagini, who becomes Voldemort's final horcrux.
Now, nowhere in the book does it say which type of snake Nagini is, which is probably what fuelled this fan theory. But it most definitely is false. Simply because Nagini is poisonous, clearly evidenced by the attack on Arthur Weasley. Now this crushes the theory quite soundly, as Boa Constrictors are not poisonous. So Nagini and the Boa Constrictor are not the same.
Nothing has been proved or denied by JKR, so we can only hope.
— Quora user Yedu Krishnan