Despite being seventh in line to the throne, the baby will be too far down the line of succession to automatically be an HRH.
King George V - Harry's great great grandfather - limited titles within the royal family in 1917.
George V declared that: "the grandchildren of the sons of any such Sovereign in the direct male line (save only the eldest living son of the eldest son of the Prince of Wales) shall have and enjoy in all occasions the style and title enjoyed by the children of Dukes of these Our Realms."
The eldest son and heir apparent of a duke can use one of his father's lesser grade peerage titles by courtesy, according to Debrett's.
So a first son of Harry's would become Earl of Dumbarton - one of the subsidiary titles Harry received from the Queen on the morning of his wedding.
A daughter would be Lady (first name) Mountbatten-Windsor, and any subsequent sons Lord (first name) Mounbatten-Windsor.
But the Queen could make changes to allow Harry and Meghan's children to be HRHs and princes and princesses.
Ahead of Prince George's birth, the monarch issued a Letters Patent to ensure the Cambridge children had fitting titles.
Without this Princess Charlotte and Prince Louis would have been a Lady and a Lord instead, but Prince George, as the eldest son of the eldest son of the Prince of Wales, would still have been a prince.
The Queen could decide to do the same for Harry and Meghan's baby.
Princess Anne’s children Zara Tindall and Peter Phillips do not have HRH status, whereas Prince Andrew’s daughters, Princess Beatrice and Eugenie do.
Zara, who was born Miss Zara Phillips, has spoken of how not having a title was a blessing.
"I've been very lucky. My parents didn't give us titles, so we've been able to have a slightly more normal upbringing. As soon as you've got a title, it's very difficult to shed it," she said.
Harry too has told of the pressures of being a prince and has always stressed the importance of being seen as 'normal'.
He confessed in 2017 that he once "wanted out" of the Royal Family.
Harry said the time he spent in the Army - when he was "just Harry" - was "the best escape I've ever had" and he once considered giving up his title.
"I felt I wanted out but then decided to stay in and work out a role for myself," he said.
The Earl and Countess of Wessex's children Lady Louise Mountbatten-Windsor and Viscount Severn are actually entitled to be a princess and prince as children of the son of the sovereign.
But the couple decided, with the Queen's agreement, that their children would use the courtesy titles as sons or daughters of an earl rather than the style prince or princess.
An example of a royal duke handing down a courtesy title to his son is the Queen's cousin the Duke of Gloucester when his son Alexander was born in 1974.
Alex is known by the duke's subsidiary title Earl of Ulster, while the duke's daughters, before their marriages, were Lady Davina Windsor, and Lady Rose Windsor.
Earl of Ulster's own son Xan - the duke's grandchild - is Lord Culleden - taken from the duke's third title Baron Culleden.
If Harry had a son who had a son - Harry's grandchild - he would be Lord Kilkeel, taken from Harry's third title Baron Kilkeel.