The family of Archie Battersbee has argued that their religious beliefs were not "given proper weight", as judges consider the case for a third time.
Since being found unconscious in his Essex home in April, 12-year-old Archie Battersbee has been in an induced coma.
His mother believes he was taking part in an online challenge.
Now, doctors want to stop treatment to protect the "dignity" of the soon-to-be teenager, but his parents are determined that he simply needs more time to recover.
As an avid boxing fan, Archie was keen to get baptised - something he'd spoken about for years.
After his accident, he was christened on Easter Sunday, whilst strapped up to the life support machine.
His mother and siblings followed suit the next day.
Despite two High Court rulings stating that Archie's life support should be removed, his mother believes only God has the right to give or take away a life.
She's believing in a miracle, and refuses to back down from a fight - the case has been taken to the Court of Appeals for a second time, in the hope of a third review of the evidence.
They argue that the religious beliefs of Archie and his family have not been taking into account.
Barrister Edward Devereux QC, who is leading Archie's parents' legal team, argued that it had been wrong to conclude that treatment was burdensome and futile.
He told appeal judges: "The overall case of the parents is that Mr Justice Hayden's decision was driven almost wholly by what is in Archie's medical best interests and not careful, clear, understandable and comprehensive evaluation of Archie's best interests in the wider sense."