Green’s Dictionary of Slang

an unspecified large number or amount, usu. used attrib.

1878		Christchurch Press 14 Sept. 5: This must be good news for the Jap. paper men, for assuming they are like you and me, and never trot around with a credit balance of more than about umteen [sic] pence, they can sling the ink of libel [...] with a sweet freedom.
1882		Northwestern Miller (Minneapolis, MN) 21 July 44: In regard to crop prospects in this section, wheat is same as ever, thanks. [...] Probable yields sufficient to ‘yank the bun’. Increase acreage ‘umpteen’’ per cent.
1915		C.E.W. Bean Anzac Book 135/2: And they counted and they found umpteen million men of valour, two ships, [...] umpteen billion rolls of paper money and ennygottsquantitee gas.
1926		Boys’ Realm 16 Jan. 265: He ran umpteen rings round them, tricked them.
c.1939		B. Bennett ‘The Street of a Thousand Lanterns’ in Billy Bennett’s Third Budget 29: They were married in umpteen what-not 2.
1940		S. Lister Mistral Hotel (1951) 173: Umpteen million tons of nickel is goin’ to get blown to hell.
1959		M. Richler Apprenticeship of Duddy Kravitz (1964) 258: Duddy could see his face so clearly and he could remember him from [...] umpteen other movies.
1968		P. Terson Apprentices (1970) I iii: How many times have I told you to keep clear of that wall? It’s a dangerous wall, and I’ve told you and told you umpteen times.
1970		A. Sillitoe Start in Life (1979) 28: Walking umpteen miles a week.
1999		Indep. Rev. 10 July 8: Author and editor of umpteen books, editions, and reference books.
2004		C. Brookmyre Be My Enemy 127: Max recounted umpteen such bafflingly nebulous posts.
2005		P. Temple Broken Shore (2007) [ebook] Charles shells out to anyone who comes along. [...] Bailed out the footy club umpteen times.
2020		D. Winslow ‘Sunset’ in Broken 186: Every Thursday for the past umpteen years has been poker night.