What is the cultural origin of "Thrice Honored"
Solution 1:
Mathematics or Hyperbole
As it has been used at least since Middle English, thrice participates in two linguistic systems of numbering. In a one, two, three system, thrice means ‘three times’:
Published to commemorate the artist/photographer's [Luis González Palma] third invitation and participation at the Venice Biennale in 2017, making him the very first and only Latin-American artist thus far to be thrice-honored by the most prestigious art retrospective in the world.
The Guatemalan photographer had been invited to the Venice Biennale three times, thus thrice-honored.
In a one, two, many system, however, thrice means ‘to a great degree’:
There was this many knyghtes that overmacched [‘overmatched’ defeated] sir Gawayne for all his thryse double myght. — Thomas Malory, Le Morte d’Arthur, ca. 1470.
Now I don't suspect Sir Thomas expected his readers to pause his tale of a noble knight to solve a simple multiplication problem, but instead is using thrice double hyperbolically. For all Gawain’s legendary prowess, there were still knights who could “overmatch” him.
Thrice Double Shakespeare, Thrise Double Harvey, Twice Jonson
Shakespeare uses thrice two times in this sense, including Malory’s thrice double:
[Orlando to Oliver] he is thrice a villain that says such a father begot villains. — AYL 1, 1, 54.
[Caliban to Prospero] What a thrice double ass / Was I... — Tem 5, 1, 296.
As a convenient intensifier, thrice invited the formation of readily understood compounds. Gabriel Harvey’s Letter-Book (1579) uses thrice — actually the older spelling thrise where the genitive ending is more transparent — with six different adjectives: happye and thrise happye (57, 92), thrise happye alone (78), thrise venerable (60), thrishonorable, in reference to Spenser’s moustache (61), thrise mightye (69), thrise dulcer (110), and thrise blessed (130).
Shakespeare uses six such compounds (or strictly thrice double, if you like):
[Tamora to Titus] Thrice-noble Titus, spare my son. Tit 1, 1, 20.
[disguised Tamora to Titus] send for Lucius, thy thrice-valiant son. — Tit 5, 2, 112.
[Ulysses to Ajax] thy parts of nature / Thrice-famed beyond ... all erudition. — TC 2, 3, 240.
[Troilus alone] What will it be, / When that the watery palate tastes indeed / Love's thrice-repured nectar? [Q; F thrice reputed] [or: three-times purified] — TC 3, 2, 20.
[Warwick to King, of Gloucester] violent hands were laid / Upon the life of this thrice-famed Duke. — 2H6, 3, 2, 157.
[Othello to all, of custom] Hath made the flinty and steel couch of war / My thrice-driven bed of down. — Oth 1, 3, 229.
Although Shakespeare does not specifically use thrice-honoured, both his plays and Malory’s romance show that one need not look to French or Latin for its origins. Both Gabriel Harvey and Shakespeare also show that the pattern could be easily followed with other adjectives to produce further compounds.
When Shakespeare’s rival Ben Jonson was briefly imprisoned in 1605 for once again running afoul of the censors, he penned a letter “to the most nobly virtuous and thrice-honoured Earl of Salisbury.” And upon Jonson’s death
Shackerley Marmion also wrote … an elegy on Jonson, published in 1638, titled “A Funeral Sacrifice, to the Sacred Memory of his Thrice-Honored Father, Ben Jonson.”
Early American Sources
As Mormion elegized Jonson, when Josiah Winslow, the first colonial governor born in America, died in 1680, a clergyman in Scituate penned an elegy in rather tortured pentameter:
Upon the much to be lamented DEATH of the thrice three times honoured JOSIAH WINSLOW, Esq., late GOVERNOUR of New Plymouth
Thrice honoured Rulers, Elders, and People all
Come and lament this stately Cedar's fall.
… Thrice Royal CHARLES, were he in person here,
Into thy Urn, would drop a sacred tear. — The Rev. William Wetherell, Scituate, Mass., 1680.
On a happier note, and with a far greater sense of the powers of English, William Penn closes a letter to his wife:
So farewell to my thrice dearly beloved wife and children! Yours, as God pleaseth, in that which no waters can quench, no time forget, nor distance wear away, but remains for ever, " Worminghurst, fourth of sixth month, 1682. William Penn. — cited in: Sydney Smith, The Edinburgh Review 21(1813), 458.
A Fourth of July speech in 1795 begins with a rhetorical flourish:
This honored, thrice honored day completes the nineteenth anniversary of our political freedom: it constitutes an august era in the annals of time, well worthy of being held in perpetual remembrance to the latest generations: it founds an illustrious epocha in the history of man, highly meriting most grateful inscription, on the living tablet of the raptured heart. — An oration on the independence of the United States of Federate America; pronounced at Portsmouth, New-Hampshire, July 4, 1795. / By George Richards.
The construction adjective, thrice adjective seems to have been a favored device, also in England:
But happy they, thrice happy, who profess Their greatest blessing is the power to bless… — The British Critic, and Quarterly Theological Review, 1801.
Welcome, thrice, welcome, ye noble patriots, to this asylum of the oppressed! — William Hamilton, Report of the Trial and Acquittal of Edward Shippen, Esq., Chief Justice, 1805.
The city of Providence, fortunate in its situation at the head-waters of Narragansett Bay, thrice fortunate in its birth of freedom and soul liberty in 1636, was most highly favored in the character and quality of its founders. — Biographical Pamphlet, 1812.
This construction apparently enjoyed a long-lived popularity. First seen in Gabriel Harvey, Phineas Fletcher manages to use happy, thrice happy three times in only seven lines:
Thrice Happy Times
Happy, thrice happy times in silver age!
…
Happy, thrice happy age! happy, thrice happy times! — Phineas Fletcher (d. 1650).
Conclusion
Thrice as a hyperbolic intensifier and its compounds offered poets and orators, especially those of lesser skills, a way of adding gravitas to their works. Thus one can imagine Coleridge's delight in addressing “thrice-honoured fleas” in a tribute to John Donne’s little erotic poem. Thrice compounds remained popular well into the twentieth century: At archive.org, a query for thrice honoured and thrice honored yielded together 2361 hits, but Fletcher’s happy, thrice happy produced 7476.
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Solution 2:
thrice Oxford Dictionaries
three times, threefold
manner or degree to a high degree‘a dose of 25 mg thrice daily’
submodifier: Extremely; very.
‘I was thrice blessed’ 3
As in: "Thrice-Honored Father", most honored.
Origin
Middle English thries, from earlier thrie (from Old English thrīga related to three)+ -s (later respelled -ce to denote the unvoiced sound); compare with once thrishonorable 1579 OED 2a
On the number 3 wikipedia
Solution 3:
I decided to answer this after seeing Hermes Trismegistus and remembering this question. In English single words that mean one time, two times ... n times only extend to three, being thrice. I'm not sure thrice-honoured is supposed to be used as "most highly honoured" because the vocabulary is limited at three. There seems to be no consensus on why "thrice", but below is at least another explanation.
What I'm leaning towards is the influence of the "thrice-great" Hermes throughout mediaeval and modern history. His work the "Hermetica" has had influence on Christian and Islamic theologians/writers. If you look at the article on Hermeticism you'll see:
Hermeticism is a religious, philosophical, and esoteric tradition based primarily upon writings attributed to Hermes Trismegistus ("Thrice Great"). These writings have greatly influenced the Western esoteric tradition and were considered to be of great importance during both the Renaissance and the Reformation. The tradition claims descent from a prisca theologia, a doctrine that affirms the existence of a single, true theology that is present in all religions and that was given by God to man in antiquity.
Many writers, including Lactantius, Cyprian of Carthage, Augustine, Marsilio Ficino, Giovanni Pico della Mirandola, Giordano Bruno, Campanella, Sir Thomas Browne, and Ralph Waldo Emerson, considered Hermes Trismegistus to be a wise pagan prophet who foresaw the coming of Christianity. St. Thomas Aquinas reported that Trismegistus arrived at something akin to the doctrine of the Trinity.
Hermeticism
So you can see in that article that in multiple times in history his ideas became associated with the Trinity.
The article also goes on to explain the influences of Hermeticism in many religions, including New Age, Neopaganism, and 20th century Esoteric Christianity.
Another explanation in that article is given from
"Hart explains that the epithet is derived from an epithet of Thoth found at the Temple of Esna, "Thoth the great, the great, the great."
Hermes Trismegistus
From The Routledge Dictionary of Egyptian Gods and Goddesses.
This is only speculation on my part. KarlG has given an ample answer with plenty of examples of its use as what he calls "a hyperbolic intensifier". As for the origin of this use, this is a possible explanation that went through my mind.