Where did the word "umpteenth" come from, and why is it a "teen", if it is supposed to represent a very large number?
Umpteenth comes from umpty, meaning an indefinite number. Etymology Online says "umpty" is derived from "Morse code slang for "dash," influenced by association with numerals such as twenty, thirty, etc."
And I think the "teenth" insert is to make the "th" easier to pronounce for the ordinal formation, but it might be also to add emphasis by way of lengthening the word. Also, it might be because of the teen/ten connection I mentioned in my comment above: an indefinite number of tens.
And this source suggests that Morse code slang word "ump" is imitative in origin, so I gather that means that that's what a dash sounds like in Morse code.
It occurred to me that this means something like _th times, then, or the equivalent of nth times (to the nth degree?)—although a single dash is Morse code for T.
Several reference books agree with the assertion that umpty originated as a Morse code term. From Robert Hendrickson, The Facts on File Encyclopedia of Word and Phrase Origins (2000):
umpteenth. Umpteenth may derive from M, or umpty, in early Morse Code, which signified a dash. By this theory, umpty came to mean "large or many" because M (umpty) was associated with the Latin M, "a thousand." Adding teen for "ten" to a shortened umpty, the result was umpteen, "many tens," meaning a very large number, and umpteenth.
From John Ayto, Dictionary of Word Origins (1990):
umpteen Umpteen was derived from an earlier umpty. This began life as a signallers' slang term for a 'dash' in Morse code (like its companion iddy for 'dot,' it was purely fanciful in origin). Its similarity to twenty, thirty, etc led to its being used for an 'indefinitely large number,' and umpteen simply replaced the -ty suffix with -teen.
From Eric Partridge, A Dictionary of Slang and Unconventional English, Fifth Edition (1961):
iddy (or itty) umpty. A signaler: military: late C. 19–20. Ex a phrase used in India for teaching Morse to native troops.
The earliest example of this usage that I could find in Google Books search results is from "An 'Iddy Umpty' Idyll," in Punch (January 24, 1906):
For the Morse Code used by the Daily Mail to announce Election results, the words "iddy" and "umpty" are commonly employed in the services in preference to "dot" and "dash."
Other references—such as Wentworth & Flexner, Dictionary of American Slang first edition (1960), and Chapman & Kipfer, Dictionary of American Slang, third edition (1995)—omit the Morse code interpretation and simply report the somewhat indefinite numbers that umpty, umpteen, and umpteenth indicate.
Wentworth & Flexner is most interesting in claiming specific numerical ranges for the terms, in at least some instances:
umpteenth [or] umteenth adj. 1 Any unspecified ordinal number from 13th to 19th inclusive. Not common. 2 Any large indeterminate ordinal number.
umpty n., adj. Any unspecified number ending in "-ty" from 20 to 90 inclusive.
umpty-umph adj. Any unspecified ordinal number from the 24th to the 99th inclusive.
A Google Books search turns up several examples of umpteen and its variant umteen from before 1908, all of them from North America. The earliest two, from a Toronto, Ontario, typographer called Griff, use the word umteen in an unusual way that I can't satisfactorily interpret. From Griff, "Toronto Topics of Interest," in The Typographical Journal (July 1, 1900):
Printers in this town are great for wheels—I mean bicycles. The latest to astonish the cinder path by wheeling is our veteran friend, John Armstrong, of the Mil. John looks quite nifty—in fact umteen—scorching along with his outfit.
From Griff, "From Toronto, Can.," in The Typographical Journal (December 15, 1901):
The newspaper writers of this city are getting together, through the efforts of Organizers Stevenson and Huddleston. Two meetings have already been held. "Watch their smoke" when they get full power on, for we think they will be "umteen" and warm members, too.
But soon after, umpteen and umteen appear in their more familiar sense. From an advertisement in Dry Goods Reporter (July 5, 1902):
But just as soon as I perfected the crackerjack I crowed. Several hundred heard me and answered—then they ordered; then I crowed again. Umpteen hundred more answered—more orders, more crowing—again more orders. Then, by jiminy! I was swanmped—swamped completely.
From a letter from Meridian, Mississippi (August 24, 1903), in Machinists' Monthly Journal (October 1903):
Now while you never read in the newspapers and scientific papers anything startling or sudden pertaining to [Lodge] 312, yet she is keeping abreast with the times in her own conservative way. In umpteen hundred and froze-to-death we were working for $2.01 per ten hours, and we are now getting close on $3.20 per day.
From "The Contributors' Club," in The Atlantic Monthly (August 1904):
Some months after, when things have "quieted down," the promoter of that institution ["The Rising-Star Ebenezer-African Mining, Development and Ameliorating Corporation, Limited"] suddenly flares forth as the "Wind and Water Promoting and Pyrotechnical Company" "controlling umteen million acres in Popotalego County, Salt River, Jumping-off-place."
From "Little Bo Peep," in [New York] Puck (July 25, 1906), credited to the Milwaukee Sentinel:
Little Bo Peep/ Lost umpteen sheep/ And she's mad as the very dickens,/ For she's got a hunch/ That all of the bunch/ Were made into potted chickens.
From "Short Talks With the Trade" in Cement World (June 15, 1907):
The owner was ready to sign the contract when along cam the clouds and the north winds that groaned umpteen kinds of bow-wows around the corners of his house till he caught a large man's size dose of grouch.
But all of these instances are latecomers compared to the use of umpty by John Seymour Wood, in College Days, or Harry's Career at Yale (1894) in various compound numbers. In this book, the different class years at Yale are consistently referred to as "Umpty-one" through "Umpty-four." Here is how a typical instance reads:
"We're here on serious business," he said with a yawn. "Delta Kappa is the largest college secret society in the United States—I mean has the largest active membership in one chapter. Why there are over seventy members in our class, sixty in Umpty-two, and about sixty in Umpty-one. That makes—how many does that make?" he asked, turning to Bob Clark.
Whether "umpteen" owes its existence to Morse code or not, it was certainly in widespread use long before 1918, the earliest date listed in Merriam-Webster's Eleventh Collegiate Dictionary (2003).
The OED says umpteen (also umteen) is from umpty (an indefinite, usually large, number) and -teen. Their first quotation is from 1918 but it was used in 19th century New Zealand and Australia.
This must be good news for the Jap. paper men, for assuming they are like you and me, and never trot round with a credit balance of more than about umteen pence, they can sling the ink of libel and wield the pen of sarcasm with a sweet freedom to which they have been hitherto strangers.
But complainant said to her he had already given her the pound. She said, ' No, that's the pound you gave me ; I know it by the number.' Complainant said ' What is the number ?' and Miss Leggett replied ' umpteen 7.' Complainant said ' That won't do, you know you put the pound I gave you in your pocket.'
… complainant said, ' Do you know the number of the note,' and she replied, ' Yes, I do ; the number is 1700 and umpteen ; she thought complainant was having a lark ; she saw complain- ant several times afterwards about the note, and he said to her,, 'I have nothing to do with it once I put it in your hands.'
He is probably by a No. 43 horse out of a No. 2 mare, or something of that sort, and most likely combines in his person the blood of "umpteen" running families and any number of sire strains.
Some other early uses from Australia:
I can answer the second part at least. Typical usage of the phrase involves the same thing happening or being done repeatedly, particularly something undesirable. For example, "This is the umpteenth time I've had to pick your socks up off the floor!"
Something undesirable happening once is bad enough. Two or three times is a bit much. If recurrences happen into the teens and beyond, at this point you are way past the realm of unique occurrences that can be excused, and are well into the realm of habitual misbehavior.