Where did the word "umpteenth" come from, and why is it a "teen", if it is supposed to represent a very large number?
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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):
From John Ayto, Dictionary of Word Origins (1990):
From Eric Partridge, A Dictionary of Slang and Unconventional English, Fifth Edition (1961):
The earliest example of this usage that I could find in Google Viewer results was this one, from the January 24, 1906, issue of Punch, in an item titled "An 'Iddy Umpty' Idyll":
Other references—such as Chapman and Kipfer's Dictionary of American Slang (1995) and Wentworth and Flexner's Dictionary of American Slang (1960)—omit the Morse code interpretation and simply report the somewhat indefinite numbers that umpty, umpteen, and umpteenth indicate.
Wentworth and Flexner are most interesting in claiming specific numerical ranges for the terms, in at least some instances:
An Ngram Viewer search turns up several examples of umpteen from before 1910.
From a letter from Meridian, Mississippi, dated August 24, 1903, to Machinists' Monthly Journal:
From Puck (July 25, 1906), credited to the Milwaukee Sentinel:
From "Short Talks With the Trade" in Cement World (June 15, 1907):
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:
Whether "umpteen" owes its existence to Morse code or not, it was certainly in widespread use long before 1918, which is the earliest date listed in Merriam-Webster's Eleventh Collegiate Dictionary (2003).
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.
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.
Some other early uses from Australia: