score:6
Simply, there would have been no Princess of Wales.
Caroline of Ansbach ceased to be the Princess of Wales on 11th June 1727 when she became Queen Consort when her husband became king.
When George II became King, his eldest son, Frederick Louis became the Prince of Wales as the title is traditionally granted to the heir apparent to the British Monarch.
When he married Augusta of Saxe-Gotha on the 17th April 1736, she automatically became the Princess of Wales. Upon her husbands death, Augusta became the Dowager Princess of Wales until her death on 8th February 1772.
The title of Princess of Wales was then not held until 8th April 1795 when Caroline of Brunswick married George, Prince of Wales who would later become King George IV.
The title Prince of Wales is traditionally only held by the Heir Apparent, and so there has not been a Princess of Wales without a Prince of Wales. With the recent changes to the law regarding succession to the throne, it may be that in the future the title may be held without a Prince of Wales.
Upvote:0
To clarify the dates a little bit:
Prince of Wales George became King George II on 22 June 1727 and his wife Caroline Princess of Wales stopped being Princess of Wales when she became Queen Caroline at the same time. Their son Frederick Lewis was created Prince of Wales on 8 January 1729. He was married om 17 April 1736 thus acquiring a Princess of Wales.
Frederick died 31 March 1751 and Princess Augusta was thereafter merely dowager Princess of Wales and not real princess of wales.
Frederick's son George was created Prince of Wales 20 April 1751. He became King George III 25 October 1760 and did not marry until 8 September 1761 and thus never had a Princess of Wales.
Upvote:16
There wasn't one.
The Princess of Wales is a courtesy title. That is, its holders are not created princesses in their own right, but rather accorded that honour only by virtue of marriage to the Prince of Wales. Accordingly, in the absence of such a prince (or when one is unmarried), there could be no Princess of Wales.
Under British law, any title to be held by the sovereign terminates. Therefore, when George Augustus ascended the throne in 1721, the title Prince of Wales was merged into the Crown and extinguished. By extension, Caroline also ceased to be Princess of Wales. The next heir apparent would have to be specifically created Prince of Wales again, by new letters patent issued by the new monarch.
To further illustrate, consider the case of Princess Augusta of Saxe-Gotha. As you noted, she became the next Princess of Wales upon marrying Prince Frederick in 1736. Prince Frederick, however, predeceased his father King George II without ever managing to ascended to the throne. The widowed Augusta was then styled Princess Dowager of Wales.
1757 book about Ernest the Pious, Duke of Saxe-Gotha and Augusta's great grandfather.