TLDR: there is pretty good evidence that George III did say privately that Washington would be the "greatest man in the world" if he retired from power after the end of the war of indepedence, which Washington subsequently did.

In 1799, the English artist Joseph Farington made the following entry in his diary:

West told me that during the American War, a sort of Committee of American Loyalists sat at New York, who had such influence with the Ministry here that their advice was followed in everything. It consisted of Andrew Eliott, a Scotchman married & settled in America, -- Franklin, -- and Joseph Galloway,-- the latter of whom came to England and was the acting adviser here. -- After the defeat of Lord Cornwallis, a report was circulated here that the Royal Standard was raised in Philadelphia. -- West was one day with the King when He came from Court to Dinner, & His Majesty mentioned the circumstances & asked West if He corresponded with any persons in America & had heard of it. West told his Majesty that a Quaker was lately arrived from Philadelphia and was with him the day before, and He asked the King when the Standard was raised. The King said the day mentioned was June 25th. -- West observed that the Quaker left Philadelphia July 1st. -- and knew nothing of such a circumstance. The Queen was present at this conversation.

The next day West had occasion to go to the Queens Palace to transact some business for the Queen, which when He had done it, she asked him if He was engaged that morning. He said not. She then told him to go into Her Closet with Her which He did & found the King sitting there. -- The King began to talk abt. America. He asked West what would Washington do were America to be declared independant. West said He believed He would retire to a private situation. -- The King said if He did He would be the greatest man in the world. He asked West how He thought the Americans would act towards this country if they became independant. West said the war had made much ill blood but that would subside, & the dispositions of many of the Chiefs, Washington, Lawrence, -- Adams, -- Franklin, -- Jay were favorable to this country which would soon have a preference to any other European Nation.

"West" is Benjamin West -- an Pennsylvania-born painter who spent his adult life in Europe, including in the service of George III. As the context makes clear, this conversation is reported to have taken place at the end of the War, after Yorktown (not after Washington had completed his presidency, as is sometimes claimed).

To clarify the chronology:

• Sometime after Yorktown, presumably 1781 or 1782, the King allegedly had this conversation with West.
• In December 1799, West allegedly recalls the saying while meeting with Farington, and Farington records it in his diary.

While one could argue West's memory of George III's exact words might not have been perfect (he was recalling a conversation almost two decades earlier), there is no compelling reason to doubt the broad veracity of the quote: Farington had no connection to the United States, and West, while born in colonial Pennsylvania, left well before independence and spent his entire adulthood in Europe hobnobbing with the elite, including the British Royal Family.

EDIT: a couple of extra notes on the chronology:

• If the diary's subsequent reference to Lord Shelburne is correct, we can date George III's remark precisely to July the 3rd, 1782.
• Washington was already dead when West reported the remark to Farington and Farington wrote it down -- but neither man could have known that. He died on on December the 14th 1799, but the news did not reach Great Britain until late January of 1800. The diary entry is dated December the 28th, 1799.