The 29-year-old, known only as "Y", was born with a genetic disorder, epilepsy, learning and development difficulties, and hearing and sight problems.
He claims the rape of his mother by her father mutated genes and raised the chance of health issues from 2% to 50%.
The Criminal Injuries Compensation Authority said Y was not a victim as he did not exist at the time.
An earlier appeal hearing ruled that Y was eligible for compensation, but the authority is now challenging that decision.
Y's mother was abused from the age of 11 by her father.  In 1991, the father - Y's grandfather - pleaded guilty to incest and was jailed for three years.
His mother won her claim under the Criminal Injuries Compensation Scheme as a victim of unlawful sexual violence.
But, when Y tried to get compensation for personal injury as a victim of the crime himself, CICA rejected his application, saying he did not exist as a person when it happened.
The First Tier Tribunal, which handles appeals, backed the CICA decision. But the case was then taken to the Upper Tribunal, which sent it back to the CICA.
Upper Tribunal judge Howard Levenson said the scheme provided for compensation to be payable to "an applicant".
"Clearly, at the time of the claim the applicant is a person," said the judge. "There is no provision in the scheme that the applicant must have been 'a person' at the time that the crime of violence was committed.
"In everyday terms and in common parlance, it seems to me that he has suffered injuries. Those injuries have been sustained in and are directly attributable to a crime of violence."
At the appeal against the Upper Tribunal's decision, CICA's counsel, Ben Collins QC, said the decision was flawed and, if the crime had not been committed, Y would not have existed.
"The harm of which Y complains was done not when Y was in the womb but in the very act of creating him," he said. "It is his own genetic make-up of which he complains."
However, Mr Collins said CICA knew the case "arises out of grave suffering on the part of Y and, in particular, his mother" and said the appeal "in no way detracts from its recognition of that."
The judges have reserved their decision and will announce it at a later date.
A severely disabled man born after an incestuous rape is seeking compensation at the Court of Appeal as a victim.