I never thought about the word, and I was a little skeptical of Wiktionary’s etymology.

umpteen

From ump (“colloquial name for the dash “—” in Morse code”) +‎ -teen.

ump

Clipping of umpire

umpire

From a Middle English rebracketing of a noumpere as an oumpere, from Old French nonper (“odd number, not even (as a tie-breaking arbitrator)”), from non (“not”) + per (“equal”), from Latin par (“equal”). Doublet of nonpareil.

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But etymonline has basically the same answer. The idea that it came directly from military slang regarding Morse code makes it more believable. Plus I’ve heard umpty used before as an arbitrarily large number, specifically as umpty odd [something].

umpteen

indefinite number, “many, a lot of,” by 1907, popularized in World War I army slang, from umpty + -teen.

umpty

1905, “of an indefinite number,” usually a large one, military slang; earlier it was Morse code slang for a dash; the form influenced by association with numerals such as twenty, thirty.