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Originally and chiefly American English, the colloquial phrase more than one can shake a stick at means: more than one can count, a considerable amount or number.

A variant of this phrase occurred, for example, in the following from an account of a baseball match, published in The Evening News (Lincoln, Nebraska, USA) of Saturday 13^th July 1895 [page 5, column 1]:

The Des Moines scored right at the start, and parenthetically it may be remarked that they had more good luck yesterday than you could shake a bundle of twigs at.

The following, on the origin of the phrase more than one can shake a stick at, is from The Word Detective, by Evan Morris, published in the Green Bay Press-Gazette (Green Bay, Wisconsin, USA) of Monday 27^th September 1999 [page D6, columns 1 & 2]:

The question, of course, is why one would be counting or measuring a crowd of something by shaking a stick at it. Shaking a stick at someone has long been considered, for good reason, a threatening gesture. There is a possibility that “more than you can shake a stick at” first arose in the context of warfare or smaller-scale hostilities, perhaps describing an overwhelmingly superior opposing force (e.g., “More Redcoats than you could shake a stick at.”).
Another possibility, and one that I find more likely, is the stick in question was shaken in the process of counting great numbers of something, perhaps used as a pointer while doing a head count of a herd of sheep or cattle. Thus, “more than you can shake a stick at” would simply mean, figuratively, “you could wave your counting stick until your arm falls off, and you still wouldn’t reach the end.”

These are, in chronological order, the earliest occurrences of the phrase more than one can shake a stick at that I have found:

1-: From The Limner, by Peter Pallet, published in the Portland Gazette, and Maine Advertiser (Portland, Maine, USA) of Monday 1^st August 1808 [page 3, column 3]—the phrase is in quotation marks, which seems to indicate that it was already in usage:

Gun Boats Ahoi!—Within a few years, Messrs. editors, various parts of the male and female dress have been distinguished by some popular name: We have had Truxton hats—Suwarrow boots—Nelson hats—Jefferson boots and shoes—Trafalgar garters—Nelson’s wave and Nelson’s ball callicoes [sic]—and more other sorts of boots, hats, ribbons, &c. than one “could shake a stick at.”

2-: From The City, published in the Lancaster Journal (Lancaster, Pennsylvania, USA) of Wednesday 5^th August 1818 [page 3, column 1]:

“Now pray, brother John, in what respect does Lancaster resemble a city?” “I’ll tell you, madam,” said my uncle, “we have in Lancaster about 15 Rookeries; a Circus and Theatre; two mineral water establishments; a professor of boot-blacking; a good many fops; as many Taverns as you can shake a stick at; and if this don’t entitle Lancaster to the appellation of City, you had better ask the Legislature to repeal the law.”

3-: From the transcript of an oration delivered in Washington, Beaufort County, North Carolina, by John Holland, published in the Alexandria Gazette & Advertiser (Alexandria, Virginia, USA) of Tuesday 6^th August 1822 [page 4, column 3]—reprinted from the Charleston City Gazette:

“Though a man may have as much money as which he can shake a stick at, there is no rael [sic] happiness unless it is with those with whom the serious contract is made.”

4-: From the Missouri Intelligencer and Boon’s Lick Advertiser (Fayette, Missouri, USA) of Friday 16^th October 1829 [page 3, column 1]:

A Prodigy!—We have received a paper called the “Western Monitor” containing no fewer than 60 misspelt words—150 typographical errors—50 dead advertisements—three scurrilous editorial articles (in the highest degree) and as many picks, monks and friars, as we could shake a stick at in a month!

5-: From a letter by ‘No Monopolist’, dated Monday 5^th October 1829, published in the Kentucky Reporter (Lexington, Kentucky, USA) of Wednesday 11^th November 1829 [page 1, column 5]—reprinted from the Petersburg Times:

Suppose my friend buys Hogs in the West, and “Fair Play” tells the public that my friend is bringing more Hogs to this market than you can shake a stick at, would this not prevent other Hog Drovers from coming to this market; in which case my friend could obtain such prices as he pleased?