With regard to early instances of "beats me" as a stand-alone phrase with a meaning along the lines of "astonishes, amazes, or perplexes me," the earliest match for the phrase that an Elephind newspaper database search finds is from "Measuring for a Supper," in the [Leesburg, Virginia] Genius of Liberty (January 9, 1830), reprinted from the New York Constellation:
A tall, raw boned, broad-backed fellow, of no very prepossessing appearance, stopped, awhile ago, at one of the hotels in Boston, and asked for supper. Schaeffer, the famous dancing master, (who, we are told, is one of the greatest wags in the country,) being present, Boniface tipped him the wink to assume, pro. tem. the duties of landlord. Schaeffer, putting on such an air of importance as became the master of the house, told the stranger he could have supper, and desired to know what he would choose.--—"Sausages," replied the other.----"Very well, sir," said the temporary landlord, stepping up to him, "I’ll take your measure, if you please." "My measure!" ejaculated the stranger, and began to draw back. "Yes, sir," continued the wag, "we always take the measure of people before we get them a meal of victuals." "What! measure a man for a meal of victuals, the same as you would for a coat or pair of trowsers?---By jingo! that beats me, I tell ye."---Then surveying his stout frame with a rueful expression of countenance, he concluded not to take supper, but content himself with a couple of crackers and a glass of cider.
An earlier instance in which "beats me" occurs in the sense of "defeats me in a game or contest" appears in an untitled item in the [Leesburg, Virginia] Genius of Liberty (December 27, 1825):
Sir Robert Walpole was fond of playing at billiards, at which his friend Dr. Monsey excelled him.—'How happens it, Monsey,' said Robert, 'that nobody beats me at billiards, or contradicts me but you?'—'The solution is easy,' said Monsey, with his usual bluntness,'l want neither place nor money from you.—Perhaps if I did, I should be as great a bungler at billiards as you are, and as submissive an auditor as any of your expecting sycophants.'
It seems quite plausible that the trajectory of usage started with physical beating (with fists, a club, or some other weapon), then moved to the figurative beating entailed in losing a sporting contest involving physical contact, then to defeat in a non-contact contest (such as billiards or cards), and finally to an admission of figurative defeat in the face of an argument or behavior that utterly flummoxes the speaker. I have no idea precisely when these proposed stages of development arose, but at least some English speakers had clearly reached the final stage by 1830.
The phrase 'beats me hollow' as a possible precursor to 'beats me'
Alternatively, "beats me" in the relevant sense may have emerged as a shortened form of the longer figurative set phrase "beats me hollow." Google Books and Hathi Trust searches turn up a number of instances of "beat[s] me hollow" from the period 1768–1810.
From John Cleland, Memoirs of a Woman of Honour, volume 1 (1768):
Upon my honour, your Lordship is a noble engineer. What a train have you laid to blow up the fort of that little spitfire? Is it for me to presume to be "necessary to your scheams of gallantry?" No, no, you beat me hollow. Till I knew your lordship, I thought I was somebody at that sport, but I find now I am nobody. If I mean to get on, I must go to school to your Lordship ; you are a perfect Matchawell at intrigues.
From "Ballad" in Airs, Ballads, &c. in the Blackamoor Wash'd White: A New Comic Opera (1776) quoted in The Monthly Review Or Literary Journal (March 1776):
Odlooks, but the tit beat me hollow, / She run me so soon off my wind, / For the more little Jerry did follow, / She left him the further behind; / But one moon-shiny night made me happy, / For home in a tiff did I jog, / And left Doll for to find a new sappy, / To dance over briar and bog.
From H.C., "An Answer to the Letter from Edmund Burke, Esq., One of the Representatives of the City of Bristol, to the Sheriffs of that City" (May 30, 1777):
In the same year [1775], Sir, was continued "An act for preventing of theft and rapine upon the northern borders of England" originally pointed by the hand of that legislature against a great number of lewd, disorderly, and lawless persons, being thieves and robbers, commonly called Moss-troopers;" and now levelled by the hand of this parliament against the young a callow rebellion which had not yet dared much beyond its northern eyry ; which (for if metaphor were but argument, you would beat me hollow) which was yet, "as it were but in the gristle, and not yet hardened into the bone of manhood."
From an unidentified poem in The Town and Country Magazine, Or, Universal Repository of Knowledge, Instruction, and Entertainment (1777):
Tho' when within the senate-house, / There I am silent as a mouse. / Yet am I patron chief of wit, / Charades are mine—a lucky hit: / That is in kindnes to the land, / I brought them o'er from Gallia's strand; / But other factors beat me hollow, / A troop of strollers soon will follow; / For fun I love, and 'tis damn'd jolly, / To see all wear the cap of folly.
From Charles Dibdin, "Ballad: In the Wags," in A Collection of Songs Selected from the Works of Mr. Dibdin, volume 3 (1792):
Now razors to grind, sweep soot ho, and all that, / Beats me hollow, and so my fine fongs I'll all tear ; There's a hole in the ballad, / I'm hoarse as a cat, / Because why, I'm depriv'd of brown Polly the fair.
From Frederick Reynolds, Life: A Comedy (1801):
Primitive. Aye, this way.—And now, as the song says, "Henceforth I'll lead a village life, / In cottage most obscure-a; / For, with this loving man and wife, / My joys are quite secure-a.
[Exeunt Primitive, Mrs. B. and Mrs. L.]
Craftly. Well, Gabriel, what do you think?
Gabriel. Think! that he beats me hollow:—I'm only a child of nature; but, damme! he's a natural. And now, if spouse undermines the stranger—
From Henry Man, "The Tour," in The Miscellaneous Works of the Late Henry Man, volume 2 (1802):
When last I wrote I did agree, / Not to describe minutiae; / But here so many things occur, / I'll speak of Chester, ere I stir; / Suppose me at the Talbot Inn, / Where this day's journal I begin: / Not like a pupil of Apollo, / Inspired rhymesters beat me hollow; / Bụt like a bard, who time bestows, / To trifle in poetic prose.
Man died in 1799, at the age of 52, so this instance is certainly from the eighteenth century.
From "Tittletattle," in Walker's Hibernian Magazine, Or, Compendium of Entertaining Knowledge (January 1806):
O had I but the whiskers, then should I be the happy man! if any of you little creatures can and will provide a remedy for my wants, haste, haste, to relieve one sadly oprest ; for at present, although twenty-two years of age, my face is as smooth and fair as my sister's. As for my dear sweet mother, she beats me hollow : she has many long hairs on her chin and face, and when she smiles or pouts they shine exactly like the sweet whiskers of a cat.
From Peter Paul Pallet, Bath Characters; or Sketches from Life (1807):
Talk of my liberties, indeed!—-what are they to the rigs and pranks of the other gemmen of the cloth? Why, there's Mr. Chip, who lives in a neighbouring village, a brother fiddle, as well as brother parson: he beats me hollow in tricks and gallantries. Two strings to his bow, my boy; two mistresses in his house at the same time—children by both; and his wife turned put of doors; and yet, Resin, he is still permitted to instruct his parishioners by his exhortations, and improve them by his example.
From John Harriott, Struggles Through Life, Exemplified in the Various Travels and Adventures in Europe, Asia, Africa, and America, of Lieut. John Harriott (1807):
I began to be tolerably expert and to receive praise for my exertions, so that in good time I might have made a tolerable good Indian. In running and climbing, the young Indians beat me hollow ; but in wrestling I was an over-match for most. In throwing, I had no kind of chance with them, but with single stick I surprized them all with the management and exercise.
From Peter Pindar, "One More Peep at the Royal Academy; or, Odes to Academicians, &c. &c.," in The Works of Peter Pindar, Esq., volume 4 (1808/1809):
Some folks declare (to wound my dove-like nature) / I mingle too much acid with my satire; / Too prone to smile at garters, stars, and strings, / And take strange liberties with Queens and Kings; / Roast on my ordeal, like an inquisition, / Peer, parson, poet, pimp, academician: / While others swear, two bastards of Apollo (The Bellman and Mathias) beat me hollow.
And from "Bad Memory," in The Monthly Mirror: Reflecting Men and Manners (December 1810):
Some of your prurient, stout, broad-backed critics have thought that my quitting the Cestus at that age [about fifty], was, like my leaving the shield at Philippi, non bene ; but I stick to my text, though they did not, in more senses of the words than one! Here too I own that his Grace [the Duke of Queensbury] beats me hollow ; and I ascribe the melancholy disadvantage, first to a custom alluded to by me in this ode, v.29, and next to our deplorable ignorance of your panaceas, or cure-alls, called, patent medicines!
The phrase "beat[s] [one] hollow," seems to have a strong tie to horse racing, as the 1776 ballad quoted above suggests. Even earlier instances of "beat hollow"—most with an explicit connection to horse racing——turn up in a Google Books search.
From "London," in the [London] General Evening Post (September 2, 1758):
The same day was a match betwixt Mr. Robinson's bay colt, got by Blaze, and Mr. Bethell's bay colt, by Cade, weight 8 st. 7 lb. one four mile heat, for 200 guineas, which was won by the latter.
Also, Lord Downe's roan filly, got by Cade, carrying 8 st. 7 lb. beat hollow Mr. Bethell's bay colt, by Starling, carrying 8 st. 10 lb. one four-mile heat, for 200 guineas.
From James Townly, High Life Below Stairs: A Farce of Two Acts, second edition (1759):
DUKE. Credit me, Baronet, they know nothing of the Turf.
Sir HARRY. I assure you, my Lord, they lost every Match ; for Crab was beat hollow, Careless threw his Rider, and Miss Slammerkin had the Distemper.
From a review of Dr. Stevens, A Practical Treatise on Fevers in The Monthly Review, Or, Literary Journal (June 1760):
The difference in the grammatical number makes the latter read like an experiment in concert, as the term is ; and as if his Pupils, at least, were present. But as there was but one pipe to the bladder, we can suppose but a single inflater at once ; whence it is clear, that if Dr. James Keil had some advantage in science, which there is a little reason to suppose, he was beat hollow in point of wind.
From "London" in The London Chronicle (October 6, 1763):
His R.H. the Duke;s Sultan beat hollow Sir J. Lowther's Wilfon's Arabian, five miles for 1000 guineas.
Mr. Vernon's Caster beat hollow Mr. Grevile's Minikin, 200 guineas.
And from James Beattie, "On the Language of Poetry," in Essays (1776):
When one hears the following lines, which abound in poetic words,
The breezy call of incense-breathing morn, / The swallow twittering from the straw-built shed, / The cock's shrill clarion, or the echoing horn, / No more shall rouse them from their lowly bed :
—one is as sensible of the dignity of the language ; as one would be of the vileness or vulgarity of that man's speech, who should prove his acquaintance with Bridewell, by interlarding his discourse with such terms as mill-doll, queer cull, or nubbing-cheat ["See the Scoundrell's Dictionary"] ; or who, in imitation of fops and gamblers, should, on the common occasions of life, talk of being beat hollow, or saving his distance ["Language of Newmarket"].
It thus appears that "beat hollow" may have originated as horse racing slang, with the meaning "defeat easily or thoroughly." But I don't know whether the original sense of the phrase was "beat like a [hollow] drum" or "beat the breath or wind out of" or some third thing. Nor can I say with confidence that "beats me" as a freestanding phrase evolved from "beat [one] hollow," although such an evolution seems not at all far-fetched.
