Dherar al-Asoussi said he had petitioned the court to also convict the men of using social media to insult the ruling Al-Sabah family.
A lower court last month sentenced the three royals, including a ruler's nephew, and two others to five years in jail for insulting top judges online.
The court also sentenced a sixth man to a one-year term and a another for 10 years in absentia. It acquitted six others.
All the men were acquitted of a charge of insulting the emir.
But Asoussi told Al-Qabas newspaper that he has challenged the rulings and demanded that all 13 defendants be given harsher jail terms of 29 years each for insulting the emir, judges and spreading false news about Kuwait.
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The appeals court has not yet set a date for the hearing. Its rulings must be verified by the supreme court.
The royals who received jail terms include Sheikh Athbi al-Fahad Al-Sabah, a nephew of the emir and the former head of the secret service police.
He is also the younger brother of international sports figure Sheikh Ahmad al-Fahad Al-Sabah, who is president of the Olympic Council of Asia and a member of the International Olympic Committee.
They remain free although Al-Qabas said police tried unsuccessfully to arrest them on at least two occasions.
The 13 people were charged with using WhatsApp and Twitter to insult the judges and publish false news accusing them of accepting bribes.
Also convicted was Sheikh Khalifa Ali Al-Sabah, the editor of Al-Watan newspaper and television which were closed by the government for violating corporate finance rules in a decision ratified by the courts.