Indefinite and fictitious numbers

Many languages have words expressing indefinite and fictitious numbers—inexact terms of indefinite size, used for comic effect, for exaggeration, as placeholder names, or when precision is unnecessary or undesirable. One technical term for such words is "non-numerical vague quantifier". Such words designed to indicate large quantities can be called "indefinite hyperbolic numerals".
Specific values used as indefinite.
Other specific numbers are occasionally used as indefinite as well. English does this with count nouns that refer to numbers: "a dozen/dozens, a score/scores, a hundred/hundreds", and similarly "thousand, million, billion". Unlike cardinal numbers, these can be pluralized, in which case they require "of" before the noun ("millions of dollars", but "five million dollars"), and require the indefinite article "a" in the singular ("a million letters" (indefinite) but "one million letters" (definite)).
Umpteen.
Umpteen, umteen or umpty is an unspecified but large number, used in a humorous fashion or to imply that it is not worth the effort to pin down the actual figure. Despite the "-teen" ending, which would seem to indicate that it lies between 12 and 20, umpteen can be much larger.
"Umpty" is first attested in 1905, in the expression "umpty-seven", implying that it is a multiple of ten. "Ump(ty)" came from a verbalization of a dash in Morse code.
"Umpteen", adding the ending "-teen", as in "thirteen", is first attested in 1918, and has become by far the most common form.
In Norwegian, "ørten" is used in a similar way, playing on the numbers from "tretten" (13) to "nitten" (19), but often signifying a much larger number.
-illion.
Words with the suffix "-illion" (e.g. zillion, gazillion, jillion, bajillion, squillion, and others) are often used as informal names for unspecified large numbers by analogy to names of large numbers such as "million" (106), "billion" (109) and t"rillion"(1012).
These words are intended to denote a number that is large enough to be unfathomable and are typically used as hyperbole or for comic effect. They have no precise value or order. They form ordinals and fractions with the usual suffix "-th", e.g. "I asked her for the jillionth time", or "-illionaire" to describe a wealthy person.
Sagan's number.
Sagan's number is the number of stars in the observable universe. It is named in honor of Carl Sagan. This number is reasonably well defined, because it is known what stars are and what the observable universe is, but its value is highly uncertain.
"Sagan's number" is to be distinguished from the sagan unit or the humorous use of the term "sagan" to denote any large quantity—specifically, any number of at least four billion, due to Sagan's association with the phrase "billions and billions".